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id="toc-Types_of_Markov_chains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Types_of_Markov_chains"> <div class="vector-toc-text"> 1.2 Types of Markov chains </div> </a> <ul id="toc-Types_of_Markov_chains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transitions"> <div class="vector-toc-text"> 1.3 Transitions </div> </a> <ul id="toc-Transitions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 3 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-A_non-Markov_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_non-Markov_example"> <div class="vector-toc-text"> 3.1 A non-Markov example </div> </a> <ul id="toc-A_non-Markov_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> 4 Formal definition </div> </a> <button aria-controls="toc-Formal_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Formal definition subsection </button> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> <li id="toc-Discrete-time_Markov_chain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete-time_Markov_chain"> <div class="vector-toc-text"> 4.1 Discrete-time Markov chain </div> </a> <ul id="toc-Discrete-time_Markov_chain-sublist" class="vector-toc-list"> <li id="toc-Variations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Variations"> <div class="vector-toc-text"> 4.1.1 Variations </div> </a> <ul id="toc-Variations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Continuous-time_Markov_chain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous-time_Markov_chain"> <div class="vector-toc-text"> 4.2 Continuous-time Markov chain </div> </a> <ul id="toc-Continuous-time_Markov_chain-sublist" class="vector-toc-list"> <li id="toc-Infinitesimal_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Infinitesimal_definition"> <div class="vector-toc-text"> 4.2.1 Infinitesimal definition </div> </a> <ul id="toc-Infinitesimal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jump_chain/holding_time_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Jump_chain/holding_time_definition"> <div class="vector-toc-text"> 4.2.2 Jump chain/holding time definition </div> </a> <ul id="toc-Jump_chain/holding_time_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transition_probability_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Transition_probability_definition"> <div class="vector-toc-text"> 4.2.3 Transition probability definition </div> </a> <ul id="toc-Transition_probability_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Finite_state_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_state_space"> <div class="vector-toc-text"> 4.3 Finite state space </div> </a> <ul id="toc-Finite_state_space-sublist" class="vector-toc-list"> <li id="toc-Stationary_distribution_relation_to_eigenvectors_and_simplices" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Stationary_distribution_relation_to_eigenvectors_and_simplices"> <div class="vector-toc-text"> 4.3.1 Stationary distribution relation to eigenvectors and simplices </div> </a> <ul id="toc-Stationary_distribution_relation_to_eigenvectors_and_simplices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time-homogeneous_Markov_chain_with_a_finite_state_space" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Time-homogeneous_Markov_chain_with_a_finite_state_space"> <div class="vector-toc-text"> 4.3.2 Time-homogeneous Markov chain with a finite state space </div> </a> <ul id="toc-Time-homogeneous_Markov_chain_with_a_finite_state_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convergence_speed_to_the_stationary_distribution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Convergence_speed_to_the_stationary_distribution"> <div class="vector-toc-text"> 4.3.3 Convergence speed to the stationary distribution </div> </a> <ul id="toc-Convergence_speed_to_the_stationary_distribution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_state_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_state_space"> <div class="vector-toc-text"> 4.4 General state space </div> </a> <ul id="toc-General_state_space-sublist" class="vector-toc-list"> <li id="toc-Harris_chains" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Harris_chains"> <div class="vector-toc-text"> 4.4.1 Harris chains </div> </a> <ul id="toc-Harris_chains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Locally_interacting_Markov_chains" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Locally_interacting_Markov_chains"> <div class="vector-toc-text"> 4.4.2 Locally interacting Markov chains </div> </a> <ul id="toc-Locally_interacting_Markov_chains-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 5 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Irreducibility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irreducibility"> <div class="vector-toc-text"> 5.1 Irreducibility </div> </a> <ul id="toc-Irreducibility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ergodicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ergodicity"> <div class="vector-toc-text"> 5.2 Ergodicity </div> </a> <ul id="toc-Ergodicity-sublist" class="vector-toc-list"> <li id="toc-Terminology" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Terminology"> <div class="vector-toc-text"> 5.2.1 Terminology </div> </a> <ul id="toc-Terminology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Index_of_primitivity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Index_of_primitivity"> <div class="vector-toc-text"> 5.2.2 Index of primitivity </div> </a> <ul id="toc-Index_of_primitivity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Measure-preserving_dynamical_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measure-preserving_dynamical_system"> <div class="vector-toc-text"> 5.3 Measure-preserving dynamical system </div> </a> <ul id="toc-Measure-preserving_dynamical_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Markovian_representations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Markovian_representations"> <div class="vector-toc-text"> 5.4 Markovian representations </div> </a> <ul id="toc-Markovian_representations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hitting_times" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hitting_times"> <div class="vector-toc-text"> 5.5 Hitting times </div> </a> <ul id="toc-Hitting_times-sublist" class="vector-toc-list"> <li id="toc-Expected_hitting_times" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Expected_hitting_times"> <div class="vector-toc-text"> 5.5.1 Expected hitting times </div> </a> <ul id="toc-Expected_hitting_times-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Time_reversal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_reversal"> <div class="vector-toc-text"> 5.6 Time reversal </div> </a> <ul id="toc-Time_reversal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Embedded_Markov_chain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Embedded_Markov_chain"> <div class="vector-toc-text"> 5.7 Embedded Markov chain </div> </a> <ul id="toc-Embedded_Markov_chain-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Special_types_of_Markov_chains" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Special_types_of_Markov_chains"> <div class="vector-toc-text"> 6 Special types of Markov chains </div> </a> <button aria-controls="toc-Special_types_of_Markov_chains-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Special types of Markov chains subsection </button> <ul id="toc-Special_types_of_Markov_chains-sublist" class="vector-toc-list"> <li id="toc-Markov_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Markov_model"> <div class="vector-toc-text"> 6.1 Markov model </div> </a> <ul id="toc-Markov_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bernoulli_scheme" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bernoulli_scheme"> <div class="vector-toc-text"> 6.2 Bernoulli scheme </div> </a> <ul id="toc-Bernoulli_scheme-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subshift_of_finite_type" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subshift_of_finite_type"> <div class="vector-toc-text"> 6.3 Subshift of finite type </div> </a> <ul id="toc-Subshift_of_finite_type-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 7 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> 7.1 Physics </div> </a> <ul id="toc-Physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chemistry"> <div class="vector-toc-text"> 7.2 Chemistry </div> </a> <ul id="toc-Chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology"> <div class="vector-toc-text"> 7.3 Biology </div> </a> <ul id="toc-Biology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Testing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Testing"> <div class="vector-toc-text"> 7.4 Testing </div> </a> <ul id="toc-Testing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solar_irradiance_variability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solar_irradiance_variability"> <div class="vector-toc-text"> 7.5 Solar irradiance variability </div> </a> <ul id="toc-Solar_irradiance_variability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Speech_recognition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Speech_recognition"> <div class="vector-toc-text"> 7.6 Speech recognition </div> </a> <ul id="toc-Speech_recognition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Information_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Information_theory"> <div class="vector-toc-text"> 7.7 Information theory </div> </a> <ul id="toc-Information_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Queueing_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Queueing_theory"> <div class="vector-toc-text"> 7.8 Queueing theory </div> </a> <ul id="toc-Queueing_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internet_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Internet_applications"> <div class="vector-toc-text"> 7.9 Internet applications </div> </a> <ul id="toc-Internet_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistics"> <div class="vector-toc-text"> 7.10 Statistics </div> </a> <ul id="toc-Statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conflict_and_combat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conflict_and_combat"> <div class="vector-toc-text"> 7.11 Conflict and combat </div> </a> <ul id="toc-Conflict_and_combat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Economics_and_finance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Economics_and_finance"> <div class="vector-toc-text"> 7.12 Economics and finance </div> </a> <ul id="toc-Economics_and_finance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Social_sciences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Social_sciences"> <div class="vector-toc-text"> 7.13 Social sciences </div> </a> <ul id="toc-Social_sciences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Music" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Music"> <div class="vector-toc-text"> 7.14 Music </div> </a> <ul id="toc-Music-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Games_and_sports" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Games_and_sports"> <div class="vector-toc-text"> 7.15 Games and sports </div> </a> <ul id="toc-Games_and_sports-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Markov_text_generators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Markov_text_generators"> <div class="vector-toc-text"> 7.16 Markov text generators </div> </a> <ul id="toc-Markov_text_generators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 10 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 11 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Markov chain</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 48 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-48" aria-hidden="true" > 48 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Markovketting" title="Markovketting – Afrikaans" lang="af" hreflang="af" data-title="Markovketting" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B3%D9%84%D8%B3%D9%84%D8%A9_%D9%85%D8%A7%D8%B1%D9%83%D9%88%D9%81" title="سلسلة ماركوف – Arabic" lang="ar" hreflang="ar" data-title="سلسلة ماركوف" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Cadena_de_M%C3%A1rkov" title="Cadena de Márkov – Asturian" lang="ast" hreflang="ast" data-title="Cadena de Márkov" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B0_%D0%B2%D0%B5%D1%80%D0%B8%D0%B3%D0%B0" title="Марковска верига – Bulgarian" lang="bg" hreflang="bg" data-title="Марковска верига" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cadena_de_M%C3%A0rkov" title="Cadena de Màrkov – Catalan" lang="ca" hreflang="ca" data-title="Cadena de Màrkov" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Markov%C5%AFv_%C5%99et%C4%9Bzec" title="Markovův řetězec – Czech" lang="cs" hreflang="cs" data-title="Markovův řetězec" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Markov-k%C3%A6de" title="Markov-kæde – Danish" lang="da" hreflang="da" data-title="Markov-kæde" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Markow-Kette" title="Markow-Kette – German" lang="de" hreflang="de" data-title="Markow-Kette" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Markovi_ahel" title="Markovi ahel – Estonian" lang="et" hreflang="et" data-title="Markovi ahel" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BB%CF%85%CF%83%CE%AF%CE%B4%CE%B1_%CE%9C%CE%AC%CF%81%CE%BA%CE%BF%CF%86" title="Αλυσίδα Μάρκοφ – Greek" lang="el" hreflang="el" data-title="Αλυσίδα Μάρκοφ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cadena_de_M%C3%A1rkov" title="Cadena de Márkov – Spanish" lang="es" hreflang="es" data-title="Cadena de Márkov" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Markov_kate" title="Markov kate – Basque" lang="eu" hreflang="eu" data-title="Markov kate" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%D9%86%D8%AC%DB%8C%D8%B1%D9%87_%D9%85%D8%A7%D8%B1%DA%A9%D9%88%D9%81" title="زنجیره مارکوف – Persian" lang="fa" hreflang="fa" data-title="زنجیره مارکوف" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Cha%C3%AEne_de_Markov" title="Chaîne de Markov – French" lang="fr" hreflang="fr" data-title="Chaîne de Markov" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Slabhra_Markov" title="Slabhra Markov – Irish" lang="ga" hreflang="ga" data-title="Slabhra Markov" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cadea_de_Markov" title="Cadea de Markov – Galician" lang="gl" hreflang="gl" data-title="Cadea de Markov" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A7%88%EB%A5%B4%EC%BD%94%ED%94%84_%EC%97%B0%EC%87%84" title="마르코프 연쇄 – Korean" lang="ko" hreflang="ko" data-title="마르코프 연쇄" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D6%80%D5%AF%D5%B8%D5%BE%D5%AB_%D5%B7%D5%B2%D5%A9%D5%A1" title="Մարկովի շղթա – Armenian" lang="hy" hreflang="hy" data-title="Մարկովի շղթա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Markovljev_lanac" title="Markovljev lanac – Croatian" lang="hr" hreflang="hr" data-title="Markovljev lanac" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rantai_Markov" title="Rantai Markov – Indonesian" lang="id" hreflang="id" data-title="Rantai Markov" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Markov-ke%C3%B0ja" title="Markov-keðja – Icelandic" lang="is" hreflang="is" data-title="Markov-keðja" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Processo_markoviano" title="Processo markoviano – Italian" lang="it" hreflang="it" data-title="Processo markoviano" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%A8%D7%A9%D7%A8%D7%AA_%D7%9E%D7%A8%D7%A7%D7%95%D7%91" title="שרשרת מרקוב – Hebrew" lang="he" hreflang="he" data-title="שרשרת מרקוב" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Markova_%C4%B7%C4%93de" title="Markova ķēde – Latvian" lang="lv" hreflang="lv" data-title="Markova ķēde" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Markovo_grandin%C4%97" title="Markovo grandinė – Lithuanian" lang="lt" hreflang="lt" data-title="Markovo grandinė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Markov-l%C3%A1nc" title="Markov-lánc – Hungarian" lang="hu" hreflang="hu" data-title="Markov-lánc" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Markovketen" title="Markovketen – Dutch" lang="nl" hreflang="nl" data-title="Markovketen" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9E%E3%83%AB%E3%82%B3%E3%83%95%E9%80%A3%E9%8E%96" title="マルコフ連鎖 – Japanese" lang="ja" hreflang="ja" data-title="マルコフ連鎖" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Markovkjede" title="Markovkjede – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Markovkjede" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Markov-prosess" title="Markov-prosess – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Markov-prosess" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%81a%C5%84cuch_Markowa" title="Łańcuch Markowa – Polish" lang="pl" hreflang="pl" data-title="Łańcuch Markowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Cadeias_de_Markov" title="Cadeias de Markov – Portuguese" lang="pt" hreflang="pt" data-title="Cadeias de Markov" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Lan%C8%9B_Markov" title="Lanț Markov – Romanian" lang="ro" hreflang="ro" data-title="Lanț Markov" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BF%D1%8C_%D0%9C%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D0%B0" title="Цепь Маркова – Russian" lang="ru" hreflang="ru" data-title="Цепь Маркова" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Zinxhir%C3%ABt_e_Markovit" title="Zinxhirët e Markovit – Albanian" lang="sq" hreflang="sq" data-title="Zinxhirët e Markovit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Markov_chain" title="Markov chain – Simple English" lang="en-simple" hreflang="en-simple" data-title="Markov chain" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BD%D1%86%D0%B8_%D0%9C%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D0%B0" title="Ланци Маркова – Serbian" lang="sr" hreflang="sr" data-title="Ланци Маркова" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Markovljev_lanac" title="Markovljev lanac – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Markovljev lanac" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Rant%C3%A9_Markov" title="Ranté Markov – Sundanese" lang="su" hreflang="su" data-title="Ranté Markov" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Markovin_ketju" title="Markovin ketju – Finnish" lang="fi" hreflang="fi" data-title="Markovin ketju" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Markovkedja" title="Markovkedja – Swedish" lang="sv" hreflang="sv" data-title="Markovkedja" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%B9%E0%B8%81%E0%B9%82%E0%B8%8B%E0%B9%88%E0%B8%A1%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B8%84%E0%B8%AD%E0%B8%9F" title="ลูกโซ่มาร์คอฟ – Thai" lang="th" hreflang="th" data-title="ลูกโซ่มาร์คอฟ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Markov_zinciri" title="Markov zinciri – Turkish" lang="tr" hreflang="tr" data-title="Markov zinciri" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BD%D1%86%D1%8E%D0%B3_%D0%9C%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D0%B0" title="Ланцюг Маркова – Ukrainian" lang="uk" hreflang="uk" data-title="Ланцюг Маркова" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%A7%D8%B1%DA%A9%D9%88%D9%88_%D8%B2%D9%86%D8%AC%DB%8C%D8%B1" title="مارکوو زنجیر – Urdu" lang="ur" hreflang="ur" data-title="مارکوو زنجیر" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/X%C3%ADch_Markov" title="Xích Markov – Vietnamese" lang="vi" hreflang="vi" data-title="Xích Markov" data-language-autonym="Tiếng Việt" 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Random process independent of past history</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Markovkate_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/260px-Markovkate_01.svg.png" decoding="async" width="260" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/390px-Markovkate_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/520px-Markovkate_01.svg.png 2x" data-file-width="563" data-file-height="563" /></a><figcaption>A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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.sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="/wiki/Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Standard_deviation_diagram_micro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability" title="Probability">Probability</a> <ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a> <ul><li><a href="/wiki/Deterministic_system" title="Deterministic system">System</a></li></ul></li> <li><a href="/wiki/Indeterminism" title="Indeterminism">Indeterminism</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li> <li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li> <li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a> <ul><li><a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li> <li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li> <li><a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li> <li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li> <li><a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a> <ul><li><a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a> <ul><li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li> <li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li> <li><a href="/wiki/Random_variable" title="Random variable">Random variable</a> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Variance" title="Variance">Variance</a></li> <li><a class="mw-selflink selflink">Markov chain</a></li> <li><a href="/wiki/Realization_(probability)" title="Realization (probability)">Observed value</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li> <li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li> <li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li> <li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li> <li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li> <li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li> <li><a href="/wiki/Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_fundamentals" title="Template:Probability fundamentals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_fundamentals" title="Template talk:Probability fundamentals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_fundamentals" title="Special:EditPage/Template:Probability fundamentals"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In probability theory and statistics, a Markov chain or Markov process is a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> describing a <a href="/wiki/Sequence" title="Sequence">sequence</a> of possible events in which the <a href="/wiki/Probability" title="Probability">probability</a> of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> sequence, in which the chain moves state at discrete time steps, gives a <a href="/wiki/Discrete-time_Markov_chain" title="Discrete-time Markov chain">discrete-time Markov chain</a> (DTMC). A <a href="/wiki/Continuous-time" class="mw-redirect" title="Continuous-time">continuous-time</a> process is called a <a href="/wiki/Continuous-time_Markov_chain" title="Continuous-time Markov chain">continuous-time Markov chain</a> (CTMC). Markov processes are named in honor of the <a href="/wiki/Russia" title="Russia">Russian</a> mathematician <a href="/wiki/Andrey_Markov" title="Andrey Markov">Andrey Markov</a>. Markov chains have many applications as <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> of real-world processes.<a href="#cite_note-MeynTweedie2009page3-1">[1]</a> They provide the basis for general stochastic simulation methods known as <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a>, which are used for simulating sampling from complex <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>, and have found application in areas including <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, <a href="/wiki/Biology" title="Biology">biology</a>, <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, <a href="/wiki/Economics" title="Economics">economics</a>, <a href="/wiki/Finance" title="Finance">finance</a>, <a href="/wiki/Information_theory" title="Information theory">information theory</a>, <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, and <a href="/wiki/Speech_processing" title="Speech processing">speech processing</a>.<a href="#cite_note-MeynTweedie2009page3-1">[1]</a><a href="#cite_note-RubinsteinKroese2011page225-2">[2]</a><a href="#cite_note-GamermanLopes2006-3">[3]</a> The adjectives Markovian and Markov are used to describe something that is related to a Markov process.<a href="#cite_note-OxfordMarkovian-4">[4]</a> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Principles">Principles</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=1" title="Edit section: Principles">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AAMarkov.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/AAMarkov.jpg/220px-AAMarkov.jpg" decoding="async" width="220" height="286" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/AAMarkov.jpg/330px-AAMarkov.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/AAMarkov.jpg/440px-AAMarkov.jpg 2x" data-file-width="954" data-file-height="1240" /></a><figcaption>Russian mathematician <a href="/wiki/Andrey_Markov" title="Andrey Markov">Andrey Markov</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> A Markov process is a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> that satisfies the <a href="/wiki/Markov_property" title="Markov property">Markov property</a> (sometimes characterized as "<a href="/wiki/Memorylessness" title="Memorylessness">memorylessness</a>"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history.<a href="#cite_note-:3-5">[5]</a> In other words, <a href="/wiki/Conditional_probability" title="Conditional probability">conditional</a> on the present state of the system, its future and past states are <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a>. A Markov chain is a type of Markov process that has either a discrete <a href="/wiki/State_space" class="mw-redirect" title="State space">state space</a> or a discrete index set (often representing time), but the precise definition of a Markov chain varies.<a href="#cite_note-Asmussen2003page73-6">[6]</a> For example, it is common to define a Markov chain as a Markov process in either <a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">discrete or continuous time</a> with a countable state space (thus regardless of the nature of time),<a href="#cite_note-Parzen1999page1882-7">[7]</a><a href="#cite_note-KarlinTaylor2012page292-8">[8]</a><a href="#cite_note-Lamperti1977chap62-9">[9]</a><a href="#cite_note-Ross1996page174and2312-10">[10]</a> but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).<a href="#cite_note-Asmussen2003page73-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Types_of_Markov_chains">Types of Markov chains</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=3" title="Edit section: Types of Markov chains">edit</a>]</div> The system's <a href="/wiki/State_space" class="mw-redirect" title="State space">state space</a> and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. continuous time: <table class="wikitable" style="width: 60%;"> <tbody><tr> <th scope="col"> </th> <th scope="col">Countable state space </th> <th scope="col">Continuous or general state space </th></tr> <tr> <th scope="row">Discrete-time </th> <td>(discrete-time) Markov chain on a countable or finite state space </td> <td><a href="/wiki/Markov_chains_on_a_measurable_state_space" title="Markov chains on a measurable state space">Markov chain on a measurable state space</a> (for example, <a href="/wiki/Harris_chain" title="Harris chain">Harris chain</a>) </td></tr> <tr> <th scope="row" style="width: 10%;">Continuous-time </th> <td style="width: 25%;">Continuous-time Markov process or Markov jump process </td> <td style="width: 25%;">Any <a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">continuous stochastic process</a> with the Markov property (for example, the <a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a>) </td></tr></tbody></table> Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain (DTMC),<a href="#cite_note-Everitt,_B.S._2002-11">[11]</a> but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention.<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a> In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see <a href="/wiki/Markov_model" title="Markov model">Markov model</a>). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the <a href="/wiki/State_space" class="mw-redirect" title="State space">state space</a> of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.<a href="#cite_note-15">[15]</a> However, many applications of Markov chains employ finite or <a href="/wiki/Countable_set" title="Countable set">countably infinite</a> state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see <a href="#Variations">Variations</a>). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise. <div class="mw-heading mw-heading3"><h3 id="Transitions">Transitions</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=4" title="Edit section: Transitions">edit</a>]</div> The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">transition matrix</a> describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are the <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> or <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>, and the random process is a mapping of these to states. The Markov property states that the <a href="/wiki/Conditional_probability_distribution" title="Conditional probability distribution">conditional probability distribution</a> for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps. Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=5" title="Edit section: History">edit</a>]</div> <a href="/wiki/Andrey_Markov" title="Andrey Markov">Andrey Markov</a> studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906.<a href="#cite_note-GrinsteadSnell1997page4643-16">[16]</a><a href="#cite_note-Bremaud2013pageIX3-17">[17]</a><a href="#cite_note-Hayes20133-18">[18]</a> Markov Processes in continuous time were discovered long before his work in the early 20th century in the form of the <a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a>.<a href="#cite_note-Ross1996page235and3583-19">[19]</a><a href="#cite_note-JarrowProtter20042-20">[20]</a><a href="#cite_note-GuttorpThorarinsdottir20122-21">[21]</a> Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with <a href="/wiki/Pavel_Nekrasov" title="Pavel Nekrasov">Pavel Nekrasov</a> who claimed independence was necessary for the <a href="/wiki/Weak_law_of_large_numbers" class="mw-redirect" title="Weak law of large numbers">weak law of large numbers</a> to hold.<a href="#cite_note-Seneta19962-22">[22]</a> In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption,<a href="#cite_note-GrinsteadSnell1997page4643-16">[16]</a><a href="#cite_note-Bremaud2013pageIX3-17">[17]</a><a href="#cite_note-Hayes20133-18">[18]</a> which had been commonly regarded as a requirement for such mathematical laws to hold.<a href="#cite_note-Hayes20133-18">[18]</a> Markov later used Markov chains to study the distribution of vowels in <a href="/wiki/Eugene_Onegin" title="Eugene Onegin">Eugene Onegin</a>, written by <a href="/wiki/Alexander_Pushkin" title="Alexander Pushkin">Alexander Pushkin</a>, and proved a <a href="/wiki/Markov_chain_central_limit_theorem" title="Markov chain central limit theorem">central limit theorem</a> for such chains.<a href="#cite_note-GrinsteadSnell1997page4643-16">[16]</a> In 1912 <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> studied Markov chains on <a href="/wiki/Finite_group" title="Finite group">finite groups</a> with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by <a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Paul</a> and <a href="/wiki/Tatyana_Ehrenfest" title="Tatyana Ehrenfest">Tatyana Ehrenfest</a> in 1907, and a branching process, introduced by <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a> and <a href="/wiki/Henry_William_Watson" title="Henry William Watson">Henry William Watson</a> in 1873, preceding the work of Markov.<a href="#cite_note-GrinsteadSnell1997page4643-16">[16]</a><a href="#cite_note-Bremaud2013pageIX3-17">[17]</a> After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by <a href="/wiki/Ir%C3%A9n%C3%A9e-Jules_Bienaym%C3%A9" title="Irénée-Jules Bienaymé">Irénée-Jules Bienaymé</a>.<a href="#cite_note-Seneta19982-23">[23]</a> Starting in 1928, <a href="/wiki/Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Maurice Fréchet</a> became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.<a href="#cite_note-GrinsteadSnell1997page4643-16">[16]</a><a href="#cite_note-BruHertz20012-24">[24]</a> <a href="/wiki/Andrei_Kolmogorov" class="mw-redirect" title="Andrei Kolmogorov">Andrey Kolmogorov</a> developed in a 1931 paper a large part of the early theory of continuous-time Markov processes.<a href="#cite_note-KendallBatchelor1990page332-25">[25]</a><a href="#cite_note-Cramer19762-26">[26]</a> Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>'s work on Einstein's model of Brownian movement.<a href="#cite_note-KendallBatchelor1990page332-25">[25]</a><a href="#cite_note-BarbutLocker2016page52-27">[27]</a> He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.<a href="#cite_note-KendallBatchelor1990page332-25">[25]</a><a href="#cite_note-Skorokhod2005page1462-28">[28]</a> Independent of Kolmogorov's work, <a href="/wiki/Sydney_Chapman_(mathematician)" title="Sydney Chapman (mathematician)">Sydney Chapman</a> derived in a 1928 paper an equation, now called the <a href="/wiki/Chapman%E2%80%93Kolmogorov_equation" title="Chapman–Kolmogorov equation">Chapman–Kolmogorov equation</a>, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.<a href="#cite_note-Bernstein20052-29">[29]</a> The differential equations are now called the Kolmogorov equations<a href="#cite_note-Anderson2012pageVII2-30">[30]</a> or the Kolmogorov–Chapman equations.<a href="#cite_note-KendallBatchelor1990page572-31">[31]</a> Other mathematicians who contributed significantly to the foundations of Markov processes include <a href="/wiki/William_Feller" title="William Feller">William Feller</a>, starting in 1930s, and then later <a href="/wiki/Eugene_Dynkin" title="Eugene Dynkin">Eugene Dynkin</a>, starting in the 1950s.<a href="#cite_note-Cramer19762-26">[26]</a> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=6" title="Edit section: Examples">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Examples_of_Markov_chains" title="Examples of Markov chains">Examples of Markov chains</a></div> <ul><li><a href="/wiki/Mark_V._Shaney" title="Mark V. Shaney">Mark V. Shaney</a> is a third-order Markov chain program, and a <a href="/wiki/Markov_text" class="mw-redirect" title="Markov text">Markov text</a> generator. It ingests the sample text (the <a href="/wiki/Tao_Te_Ching" title="Tao Te Ching">Tao Te Ching</a>, or the posts of a <a href="/wiki/Usenet" title="Usenet">Usenet</a> group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those two in one of the triplets in its massive list. If there is more than one, it picks at random (identical triplets count separately, so a sequence which occurs twice is twice as likely to be picked as one which only occurs once). It then adds that word to the generated text. Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on.<a href="#cite_note-curious-32">[32]</a></li></ul> <ul><li><a href="/wiki/Random_walk" title="Random walk">Random walks</a> based on integers and the <a href="/wiki/Gambler%27s_ruin" title="Gambler's ruin">gambler's ruin</a> problem are examples of Markov processes.<a href="#cite_note-Florescu2014page3732-33">[33]</a><a href="#cite_note-KarlinTaylor2012page492-34">[34]</a> Some variations of these processes were studied hundreds of years earlier in the context of independent variables.<a href="#cite_note-Weiss2006page12-35">[35]</a><a href="#cite_note-Shlesinger1985page82-36">[36]</a> Two important examples of Markov processes are the <a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a>, also known as the <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> process, and the <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>,<a href="#cite_note-Ross1996page235and3583-19">[19]</a> which are considered the most important and central stochastic processes in the theory of stochastic processes.<a href="#cite_note-Parzen19992-37">[37]</a><a href="#cite_note-doob1953stochasticP46to472-38">[38]</a><a href="#cite_note-39">[39]</a> These two processes are Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.<a href="#cite_note-Florescu2014page3732-33">[33]</a><a href="#cite_note-KarlinTaylor2012page492-34">[34]</a></li> <li>A famous Markov chain is the so-called "drunkard's walk", a random walk on the <a href="/wiki/Number_line" title="Number line">number line</a> where, at each step, the position may change by +1 or −1 with equal probability. From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6.</li> <li>A series of independent states (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next state depends on the current one.</li></ul> <div class="mw-heading mw-heading3"><h3 id="A_non-Markov_example">A non-Markov example</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=7" title="Edit section: A non-Markov example">edit</a>]</div> Suppose that there is a coin purse containing five coins worth 25¢, five coins worth 10¢ and five coins worth 5¢, and one by one, coins are randomly drawn from the purse and are set on a table. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"> represents the total value of the coins set on the table after n draws, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a341262173c747b2994e894e171b7c9b2a718c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.509ex;" alt="{\displaystyle X_{0}=0}">, then the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{n}:n\in \mathbb {N} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{n}:n\in \mathbb {N} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b12736279654bc1369a7620cdf4d9207ad79373b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:2.843ex;" alt="{\displaystyle \{X_{n}:n\in \mathbb {N} \}}"> is not a Markov process. To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{6}=\$0.50}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">$</mi> <mn>0.50</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{6}=\$0.50}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b9122157c685c063e4d515319a94c9b438399f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.374ex; height:2.676ex;" alt="{\displaystyle X_{6}=\$0.50}">. If we know not just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489d3e6151450285af53210948fbb0706eb99711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{6}}">, but the earlier values as well, then we can determine which coins have been drawn, and we know that the next coin will not be a nickel; so we can determine that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{7}\geq \$0.60}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>≥</mo> <mi mathvariant="normal">$</mi> <mn>0.60</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{7}\geq \$0.60}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e25385671958295f24891b0623104212a6cc7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.374ex; height:2.676ex;" alt="{\displaystyle X_{7}\geq \$0.60}"> with probability 1. But if we do not know the earlier values, then based only on the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489d3e6151450285af53210948fbb0706eb99711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{6}}"> we might guess that we had drawn four dimes and two nickels, in which case it would certainly be possible to draw another nickel next. Thus, our guesses about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{7}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7d9af33b651d172949cee5952f57dbeb662eaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{7}}"> are impacted by our knowledge of values prior to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489d3e6151450285af53210948fbb0706eb99711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{6}}">. However, it is possible to model this scenario as a Markov process. Instead of defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"> to represent the total value of the coins on the table, we could define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"> to represent the count of the various coin types on the table. For instance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{6}=1,0,5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{6}=1,0,5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6faa374211d9eb484609347da795b18a200b53c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.632ex; height:2.509ex;" alt="{\displaystyle X_{6}=1,0,5}"> could be defined to represent the state where there is one quarter, zero dimes, and five nickels on the table after 6 one-by-one draws. This new model could be represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\times 6\times 6=216}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>×</mo> <mn>6</mn> <mo>×</mo> <mn>6</mn> <mo>=</mo> <mn>216</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6\times 6\times 6=216}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21a5c79acd0881855a171386ac006f74b3bd5dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.754ex; height:2.176ex;" alt="{\displaystyle 6\times 6\times 6=216}"> possible states, where each state represents the number of coins of each type (from 0 to 5) that are on the table. (Not all of these states are reachable within 6 draws.) Suppose that the first draw results in state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}=0,1,0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}=0,1,0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7afaaa321a99a7abc56e717fcbc81355f4f51b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.632ex; height:2.509ex;" alt="{\displaystyle X_{1}=0,1,0}">. The probability of achieving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}"> now depends on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}">; for example, the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}=1,0,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}=1,0,1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550dacc66eb170afd105ce93e5bd3b00d464b365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.632ex; height:2.509ex;" alt="{\displaystyle X_{2}=1,0,1}"> is not possible. After the second draw, the third draw depends on which coins have so far been drawn, but no longer only on the coins that were drawn for the first state (since probabilistically important information has since been added to the scenario). In this way, the likelihood of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}=i,j,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}=i,j,k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ea04d7ee218c31af9f7ff448da0f19b0650f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.281ex; height:2.509ex;" alt="{\displaystyle X_{n}=i,j,k}"> state depends exclusively on the outcome of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n-1}=\ell ,m,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>ℓ</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n-1}=\ell ,m,p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710d259e6ae2577a95bbf0d30ff479bd1532e665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.589ex; height:2.509ex;" alt="{\displaystyle X_{n-1}=\ell ,m,p}"> state. <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=8" title="Edit section: Formal definition">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Discrete-time_Markov_chain">Discrete-time Markov chain</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=9" title="Edit section: Discrete-time Markov chain">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Discrete-time_Markov_chain" title="Discrete-time Markov chain">Discrete-time Markov chain</a></div> A discrete-time Markov chain is a sequence of <a href="/wiki/Random_variable" title="Random variable">random variables</a> X1, X2, X3, ... with the <a href="/wiki/Markov_property" title="Markov property">Markov property</a>, namely that the probability of moving to the next state depends only on the present state and not on the previous states: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ea39e3455c78fb327846f28a47c1c45aef6719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.283ex; height:2.843ex;" alt="{\displaystyle \Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),}"> if both <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probabilities</a> are well defined, that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adfb2f8cf35951e87876921ade3f8ffa28262125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.64ex; height:2.843ex;" alt="{\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.}"></dd></dl> The possible values of Xi form a <a href="/wiki/Countable_set" title="Countable set">countable set</a> S called the state space of the chain. <div class="mw-heading mw-heading4"><h4 id="Variations">Variations</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=10" title="Edit section: Variations">edit</a>]</div> <ul><li>Time-homogeneous Markov chains are processes where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfd22ce3cac1673c52b44fad057e583d18d896d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.717ex; height:2.843ex;" alt="{\displaystyle \Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)}"> for all n. The probability of the transition is independent of n.</li> <li>Stationary Markov chains are processes where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X_{0}=x_{0},X_{1}=x_{1},\ldots ,X_{k}=x_{k})=\Pr(X_{n}=x_{0},X_{n+1}=x_{1},\ldots ,X_{n+k}=x_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{0}=x_{0},X_{1}=x_{1},\ldots ,X_{k}=x_{k})=\Pr(X_{n}=x_{0},X_{n+1}=x_{1},\ldots ,X_{n+k}=x_{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/373a86ddefbb26d2bb13c8ca33736d9cb406c8b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.727ex; height:2.843ex;" alt="{\displaystyle \Pr(X_{0}=x_{0},X_{1}=x_{1},\ldots ,X_{k}=x_{k})=\Pr(X_{n}=x_{0},X_{n+1}=x_{1},\ldots ,X_{n+k}=x_{k})}"> for all n and k. Every stationary chain can be proved to be time-homogeneous by Bayes' rule.<div class="paragraphbreak" style="margin-top:0.5em"></div>A necessary and sufficient condition for a time-homogeneous Markov chain to be stationary is that the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6381fdad2b9f11954b1fc2db08bbaccf634ededa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}"> is a stationary distribution of the Markov chain.</li> <li>A Markov chain with memory (or a Markov chain of order m) where m is finite, is a process satisfying <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{}&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{1}=x_{1})\\=&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{n-m}=x_{n-m}){\text{ for }}n>m\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>n</mi> <mo>></mo> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{1}=x_{1})\\=&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{n-m}=x_{n-m}){\text{ for }}n>m\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a66a7745730f8d414d000265010e9cd75605891" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.883ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}{}&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{1}=x_{1})\\=&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{n-m}=x_{n-m}){\text{ for }}n>m\end{aligned}}}"> In other words, the future state depends on the past m states. It is possible to construct a chain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41ef15e7a468d1c14dfb0f533a72adaa05ffe2a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle (Y_{n})}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3076fe5f4da5c004de5c8a5f99eb0f8b2989f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle (X_{n})}"> which has the 'classical' Markov property by taking as state space the ordered m-tuples of X values, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{n}=\left(X_{n},X_{n-1},\ldots ,X_{n-m+1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{n}=\left(X_{n},X_{n-1},\ldots ,X_{n-m+1}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e6a7392137ce69bdef325f848432f78bf0b185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.04ex; height:2.843ex;" alt="{\displaystyle Y_{n}=\left(X_{n},X_{n-1},\ldots ,X_{n-m+1}\right)}">.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Continuous-time_Markov_chain">Continuous-time Markov chain</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=11" title="Edit section: Continuous-time Markov chain">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Continuous-time_Markov_chain" title="Continuous-time Markov chain">Continuous-time Markov chain</a></div> A continuous-time Markov chain (Xt)t ≥ 0 is defined by a finite or countable state space S, a <a href="/wiki/Transition_rate_matrix" class="mw-redirect" title="Transition rate matrix">transition rate matrix</a> Q with dimensions equal to that of the state space and initial probability distribution defined on the state space. For i ≠ j, the elements qij are non-negative and describe the rate of the process transitions from state i to state j. The elements qii are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one. There are three equivalent definitions of the process.<a href="#cite_note-norris1-40">[40]</a> <div class="mw-heading mw-heading4"><h4 id="Infinitesimal_definition">Infinitesimal definition</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=12" title="Edit section: Infinitesimal definition">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Intensities_vs_transition_probabilities.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Intensities_vs_transition_probabilities.svg/220px-Intensities_vs_transition_probabilities.svg.png" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Intensities_vs_transition_probabilities.svg/330px-Intensities_vs_transition_probabilities.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Intensities_vs_transition_probabilities.svg/440px-Intensities_vs_transition_probabilities.svg.png 2x" data-file-width="297" data-file-height="227" /></a><figcaption>The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j.</figcaption></figure> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}"> be the random variable describing the state of the process at time t, and assume the process is in a state i at time t. Then, knowing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}=i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260dbfdd71c13647b53b97b7871750d6ca3b9d8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.651ex; height:2.509ex;" alt="{\displaystyle X_{t}=i}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t+h}=j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>+</mo> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t+h}=j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf2535616cbe38a21566632818a8af76ab00253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.032ex; height:2.509ex;" alt="{\displaystyle X_{t+h}=j}"> is independent of previous values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X_{s}:s<t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>:</mo> <mi>s</mi> <mo><</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(X_{s}:s<t\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a95f5c63111a32db00d348f62f2e0732ce4b515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.703ex; height:2.843ex;" alt="{\displaystyle \left(X_{s}:s<t\right)}">, and as h → 0 for all j and for all t, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>j</mi> <mo>∣</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi>h</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5170649f2f530ea1da7a32561ec464660deb7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.7ex; height:3.009ex;" alt="{\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}"> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, using the <a href="/wiki/Little-o_notation" class="mw-redirect" title="Little-o notation">little-o notation</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b08ec83005828a8789b639e4944b11905e9b18b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.514ex; height:2.343ex;" alt="{\displaystyle q_{ij}}"> can be seen as measuring how quickly the transition from i to j happens. <div class="mw-heading mw-heading4"><h4 id="Jump_chain/holding_time_definition">Jump chain/holding time definition</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=13" title="Edit section: Jump chain/holding time definition">edit</a>]</div> Define a discrete-time Markov chain Yn to describe the nth jump of the process and variables S1, S2, S3, ... to describe holding times in each of the states where Si follows the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a> with rate parameter −qYiYi. <div class="mw-heading mw-heading4"><h4 id="Transition_probability_definition">Transition probability definition</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=14" title="Edit section: Transition probability definition">edit</a>]</div> For any value n = 0, 1, 2, 3, ... and times indexed up to this value of n: t0, t1, t2, ... and all states recorded at these times i0, i1, i2, i3, ... it holds that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(X_{t_{n+1}}=i_{n+1}\mid X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots ,X_{t_{n}}=i_{n})=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{t_{n+1}}=i_{n+1}\mid X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots ,X_{t_{n}}=i_{n})=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e04f417634aec6a189ef76bce9750e024b3ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:71.138ex; height:3.009ex;" alt="{\displaystyle \Pr(X_{t_{n+1}}=i_{n+1}\mid X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots ,X_{t_{n}}=i_{n})=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})}"></dd></dl> where pij is the solution of the <a href="/wiki/Forward_equation" class="mw-redirect" title="Forward equation">forward equation</a> (a <a href="/wiki/First-order_differential_equation" class="mw-redirect" title="First-order differential equation">first-order differential equation</a>) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'(t)=P(t)Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P'(t)=P(t)Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0b7c78528002a7f0431a9ee0b076de7e653a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.486ex; height:3.009ex;" alt="{\displaystyle P'(t)=P(t)Q}"></dd></dl> with initial condition P(0) is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. <div class="mw-heading mw-heading3"><h3 id="Finite_state_space">Finite state space</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=15" title="Edit section: Finite state space">edit</a>]</div> If the state space is <a href="/wiki/Finite_set" title="Finite set">finite</a>, the transition probability distribution can be represented by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>, called the transition matrix, with the (i, j)th <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">element</a> of P equal to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>j</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48cd8ca68181ff3477fe5a596e83c58e14133979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:29.066ex; height:3.009ex;" alt="{\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).}"></dd></dl> Since each row of P sums to one and all elements are non-negative, P is a <a href="/wiki/Right_stochastic_matrix" class="mw-redirect" title="Right stochastic matrix">right stochastic matrix</a>. <div class="mw-heading mw-heading4"><h4 id="Stationary_distribution_relation_to_eigenvectors_and_simplices">Stationary distribution relation to eigenvectors and simplices</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=16" title="Edit section: Stationary distribution relation to eigenvectors and simplices">edit</a>]</div> A stationary distribution π is a (row) vector, whose entries are non-negative and sum to 1, is unchanged by the operation of transition matrix P on it and so is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbf {P} =\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbf {P} =\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fcd3f732d72ead24fd57a7e070bd646c5cd023c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.236ex; height:2.176ex;" alt="{\displaystyle \pi \mathbf {P} =\pi .}"></dd></dl> By comparing this definition with that of an <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a> we see that the two concepts are related and that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={\frac {e}{\sum _{i}{e_{i}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={\frac {e}{\sum _{i}{e_{i}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf84e20cfe4b267c737552ca7c67f1b996522d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.79ex; height:5.509ex;" alt="{\displaystyle \pi ={\frac {e}{\sum _{i}{e_{i}}}}}"></dd></dl> is a normalized (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}\pi _{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}\pi _{i}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890e3ec337b57632ae123e8c381865b6d5ac9c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.026ex; height:3.009ex;" alt="{\textstyle \sum _{i}\pi _{i}=1}">) multiple of a left eigenvector e of the transition matrix P with an <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> of 1. If there is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state. But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution. The values of a stationary distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \pi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \pi _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1253fb28e6c9de8aed0b2dc92cf30b15018e2e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.009ex;" alt="{\displaystyle \textstyle \pi _{i}}"> are associated with the state space of P and its eigenvectors have their relative proportions preserved. Since the components of π are positive and the constraint that their sum is unity can be rewritten as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}1\cdot \pi _{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mn>1</mn> <mo>⋅</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}1\cdot \pi _{i}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ffc271d2437f848d0a2e6b0d1a99a1ac4f8942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.868ex; height:3.009ex;" alt="{\textstyle \sum _{i}1\cdot \pi _{i}=1}"> we see that the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of π with a vector whose components are all 1 is unity and that π lies on a <a href="/wiki/Standard_simplex" class="mw-redirect" title="Standard simplex">simplex</a>. <div class="mw-heading mw-heading4"><h4 id="Time-homogeneous_Markov_chain_with_a_finite_state_space">Time-homogeneous Markov chain with a finite state space</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=17" title="Edit section: Time-homogeneous Markov chain with a finite state space">edit</a>]</div> If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th power of the transition matrix, Pk. If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π.<a href="#cite_note-auto-41">[41]</a> Additionally, in this case Pk converges to a rank-one matrix in which each row is the stationary distribution π: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30fef6ae8cf5c557c3d14832141905fec51852ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.212ex; height:4.343ex;" alt="{\displaystyle \lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }"></dd></dl> where 1 is the column vector with all entries equal to 1. This is stated by the <a href="/wiki/Perron%E2%80%93Frobenius_theorem" title="Perron–Frobenius theorem">Perron–Frobenius theorem</a>. If, by whatever means, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1678070bc1843d9b56763ce460ebfbf2f50a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.907ex; height:3.009ex;" alt="{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}"> is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below. For some stochastic matrices P, the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1678070bc1843d9b56763ce460ebfbf2f50a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.907ex; height:3.009ex;" alt="{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}"> does not exist while the stationary distribution does, as shown by this example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="2em" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mspace width="2em" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379f7b384d240fe10ec46c5af6ec0d05e05fac4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.806ex; height:6.176ex;" alt="{\displaystyle \mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P} }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb4ffc39311bdacd5ebba3e226e446e58152fce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.959ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}"></dd></dl> (This example illustrates a periodic Markov chain.) Because there are a number of different special cases to consider, the process of finding this limit if it exists can be a lengthy task. However, there are many techniques that can assist in finding this limit. Let P be an n×n matrix, and define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/740df87a293e778dfe7e76ee18a0501e09f6c8a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.66ex; height:3.009ex;" alt="{\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}"> It is always true that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {QP} =\mathbf {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {QP} =\mathbf {Q} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0e4a16aadd8e341b3e91ad532cfab38329efcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.588ex; height:2.509ex;" alt="{\displaystyle \mathbf {QP} =\mathbf {Q} .}"></dd></dl> Subtracting Q from both sides and factoring then yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6261c75de4254f32f9174766a44076d950391ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.461ex; height:3.009ex;" alt="{\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}"></dd></dl> where In is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> of size n, and 0n,n is the <a href="/wiki/Zero_matrix" title="Zero matrix">zero matrix</a> of size n×n. Multiplying together stochastic matrices always yields another stochastic matrix, so Q must be a <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">stochastic matrix</a> (see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that Q is a stochastic matrix to solve for Q. Including the fact that the sum of each the rows in P is 1, there are n+1 equations for determining n unknowns, so it is computationally easier if on the one hand one selects one row in Q and substitutes each of its elements by one, and on the other one substitutes the corresponding element (the one in the same column) in the vector 0, and next left-multiplies this latter vector by the inverse of transformed former matrix to find Q. Here is one method for doing so: first, define the function f(A) to return the matrix A with its right-most column replaced with all 1's. If [f(P − In)]−1 exists then<a href="#cite_note-42">[42]</a><a href="#cite_note-auto-41">[41]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})[f(\mathbf {P} -\mathbf {I} _{n})]^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})[f(\mathbf {P} -\mathbf {I} _{n})]^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b90f245a33e8814743118543f81825ce084a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.454ex; height:3.343ex;" alt="{\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})[f(\mathbf {P} -\mathbf {I} _{n})]^{-1}.}"></dd> <dd>Explain: The original matrix equation is equivalent to a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of n×n linear equations</a> in n×n variables. And there are n more linear equations from the fact that Q is a right <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">stochastic matrix</a> whose each row sums to 1. So it needs any n×n independent linear equations of the (n×n+n) equations to solve for the n×n variables. In this example, the n equations from "Q multiplied by the right-most column of (P-In)" have been replaced by the n stochastic ones.</dd></dl> One thing to notice is that if P has an element Pi,i on its main diagonal that is equal to 1 and the ith row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers Pk. Hence, the ith row or column of Q will have the 1 and the 0's in the same positions as in P. <div class="mw-heading mw-heading4"><h4 id="Convergence_speed_to_the_stationary_distribution">Convergence speed to the stationary distribution</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=18" title="Edit section: Convergence speed to the stationary distribution">edit</a>]</div> As stated earlier, from the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b63e21be930807aaa9235eafa1bc35bbd06277e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.742ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,}"> (if exists) the stationary (or steady state) distribution π is a left eigenvector of row <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">stochastic matrix</a> P. Then assuming that P is diagonalizable or equivalently that P has n linearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is, <a href="/wiki/Defective_matrix" title="Defective matrix">defective matrices</a>, one may start with the <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a> of P and proceed with a bit more involved set of arguments in a similar way.<a href="#cite_note-43">[43]</a>) Let U be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector of P and let Σ be the diagonal matrix of left eigenvalues of P, that is, Σ = diag(λ1,λ2,λ3,...,λn). Then by <a href="/wiki/Eigendecomposition" class="mw-redirect" title="Eigendecomposition">eigendecomposition</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">Σ</mi> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31484acca8c2eee96118667cb93fb6e1f7f30cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.949ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}"></dd></dl> Let the eigenvalues be enumerated such that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mo>⋯</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ae9bc345ea368a03d5cbceb537e12ce3852590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.002ex; height:2.843ex;" alt="{\displaystyle 1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.}"></dd></dl> Since P is a row stochastic matrix, its largest left eigenvalue is 1. If there is a unique stationary distribution, then the largest eigenvalue and the corresponding eigenvector is unique too (because there is no other π which solves the stationary distribution equation above). Let ui be the i-th column of U matrix, that is, ui is the left eigenvector of P corresponding to λi. Also let x be a length n row vector that represents a valid probability distribution; since the eigenvectors ui span <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"> we can write <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="2em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9538effd1b72ff6b73892661b1a667c026eeb553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.792ex; height:6.843ex;" alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R} .}"></dd></dl> If we multiply x with P from right and continue this operation with the results, in the end we get the stationary distribution π. In other words, π = a1 u1 ← xPP...P = xPk as k → ∞. That means <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {\pi }}^{(k)}&=\mathbf {x} \left(\mathbf {U\Sigma U} ^{-1}\right)\left(\mathbf {U\Sigma U} ^{-1}\right)\cdots \left(\mathbf {U\Sigma U} ^{-1}\right)\\&=\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}\\&=\left(a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\mathbf {u} _{n}^{\mathsf {T}}\right)\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1}\\&=a_{1}\lambda _{1}^{k}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\lambda _{n}^{k}\mathbf {u} _{n}^{\mathsf {T}}&&u_{i}\bot u_{j}{\text{ for }}i\neq j\\&=\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}^{\mathsf {T}}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}^{\mathsf {T}}+\cdots +a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}^{\mathsf {T}}\right\}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">π</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">Σ</mi> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">Σ</mi> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">Σ</mi> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">Σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> </mtd> <mtd /> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">⊥</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mrow> <mo>{</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msubsup> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {\pi }}^{(k)}&=\mathbf {x} \left(\mathbf {U\Sigma U} ^{-1}\right)\left(\mathbf {U\Sigma U} ^{-1}\right)\cdots \left(\mathbf {U\Sigma U} ^{-1}\right)\\&=\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}\\&=\left(a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\mathbf {u} _{n}^{\mathsf {T}}\right)\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1}\\&=a_{1}\lambda _{1}^{k}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\lambda _{n}^{k}\mathbf {u} _{n}^{\mathsf {T}}&&u_{i}\bot u_{j}{\text{ for }}i\neq j\\&=\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}^{\mathsf {T}}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}^{\mathsf {T}}+\cdots +a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}^{\mathsf {T}}\right\}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbece5b0ebe47aed648fdf4dc5debdb0d9779bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; width:92.172ex; height:21.509ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\pi }}^{(k)}&=\mathbf {x} \left(\mathbf {U\Sigma U} ^{-1}\right)\left(\mathbf {U\Sigma U} ^{-1}\right)\cdots \left(\mathbf {U\Sigma U} ^{-1}\right)\\&=\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}\\&=\left(a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\mathbf {u} _{n}^{\mathsf {T}}\right)\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1}\\&=a_{1}\lambda _{1}^{k}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\lambda _{n}^{k}\mathbf {u} _{n}^{\mathsf {T}}&&u_{i}\bot u_{j}{\text{ for }}i\neq j\\&=\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}^{\mathsf {T}}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}^{\mathsf {T}}+\cdots +a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}^{\mathsf {T}}\right\}\end{aligned}}}"></dd></dl> Since π is parallel to u1(normalized by L2 norm) and π(k) is a probability vector, π(k) approaches to a1 u1 = π as k → ∞ with a speed in the order of λ2/λ1 exponentially. This follows because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mo>⋯</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da30b4e1c34fb872eeb742aecd57f078f30d2851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.138ex; height:2.843ex;" alt="{\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,}"> hence λ2/λ1 is the dominant term. The smaller the ratio is, the faster the convergence is.<a href="#cite_note-44">[44]</a> Random noise in the state distribution π can also speed up this convergence to the stationary distribution.<a href="#cite_note-45">[45]</a> <div class="mw-heading mw-heading3"><h3 id="General_state_space">General state space</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=19" title="Edit section: General state space">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Markov_chains_on_a_measurable_state_space" title="Markov chains on a measurable state space">Markov chains on a measurable state space</a></div> <div class="mw-heading mw-heading4"><h4 id="Harris_chains">Harris chains</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=20" title="Edit section: Harris chains">edit</a>]</div> Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through <a href="/wiki/Harris_chain" title="Harris chain">Harris chains</a>. The use of Markov chains in Markov chain Monte Carlo methods covers cases where the process follows a continuous state space. <div class="mw-heading mw-heading4"><h4 id="Locally_interacting_Markov_chains">Locally interacting Markov chains</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=21" title="Edit section: Locally interacting Markov chains">edit</a>]</div> "Locally interacting Markov chains" are Markov chains with an evolution that takes into account the state of other Markov chains. This corresponds to the situation when the state space has a (Cartesian-) product form. See <a href="/wiki/Interacting_particle_system" title="Interacting particle system">interacting particle system</a> and <a href="/wiki/Stochastic_cellular_automata" class="mw-redirect" title="Stochastic cellular automata">stochastic cellular automata</a> (probabilistic cellular automata). See for instance Interaction of Markov Processes<a href="#cite_note-46">[46]</a> or.<a href="#cite_note-47">[47]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=22" title="Edit section: Properties">edit</a>]</div> Two states are said to communicate with each other if both are reachable from one another by a sequence of transitions that have positive probability. This is an equivalence relation which yields a set of communicating classes. A class is closed if the probability of leaving the class is zero. A Markov chain is irreducible if there is one communicating class, the state space. A state i has period k if k is the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of the number of transitions by which i can be reached, starting from i. That is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>></mo> <mn>0</mn> <mo>:</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo>∣</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1a7f9a343bd02a6319adb3036e6bd20aaf6be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.14ex; height:2.843ex;" alt="{\displaystyle k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}}"></dd></dl> The state is periodic if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda43bd4034dc2d04cd562005d0af81d3d2dbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>1}">; otherwise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"> and the state is aperiodic. A state i is said to be transient if, starting from i, there is a non-zero probability that the chain will never return to i. It is called recurrent (or persistent) otherwise.<a href="#cite_note-Heyman-48">[48]</a> For a recurrent state i, the mean hitting time is defined as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}=E[T_{i}]=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> <mo>⋅</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}=E[T_{i}]=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a59758d6b3e435f8b501e513653d1387277ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.758ex; height:6.843ex;" alt="{\displaystyle M_{i}=E[T_{i}]=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}.}"></dd></dl> State i is positive recurrent if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}"> is finite and null recurrent otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties — that is, if one state has the property then all states in its communicating class have the property.<a href="#cite_note-49">[49]</a> A state i is called absorbing if there are no outgoing transitions from the state. <div class="mw-heading mw-heading3"><h3 id="Irreducibility">Irreducibility</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=23" title="Edit section: Irreducibility">edit</a>]</div> Since periodicity is a class property, if a Markov chain is irreducible, then all its states have the same period. In particular, if one state is aperiodic, then the whole Markov chain is aperiodic.<a href="#cite_note-50">[50]</a> If a finite Markov chain is irreducible, then all states are positive recurrent, and it has a unique stationary distribution given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{i}=1/E[T_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{i}=1/E[T_{i}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd613a6003008f50be231e1b7c27299bdf969db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.775ex; height:2.843ex;" alt="{\displaystyle \pi _{i}=1/E[T_{i}]}">. <div class="mw-heading mw-heading3"><h3 id="Ergodicity">Ergodicity</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=24" title="Edit section: Ergodicity">edit</a>]</div> A state i is said to be ergodic if it is aperiodic and positive recurrent. In other words, a state i is ergodic if it is recurrent, has a period of 1, and has finite mean recurrence time. If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. Equivalently, there exists some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> such that all entries of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd66ef6f4af0b81e278ea2a32de5d5d7d3ea6613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.587ex; height:2.676ex;" alt="{\displaystyle M^{k}}"> are positive. It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. More generally, a Markov chain is ergodic if there is a number N such that any state can be reached from any other state in any number of steps less or equal to a number N. In case of a fully connected transition matrix, where all transitions have a non-zero probability, this condition is fulfilled with N = 1. A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic. <div class="mw-heading mw-heading4"><h4 id="Terminology">Terminology</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=25" title="Edit section: Terminology">edit</a>]</div> Some authors call any irreducible, positive recurrent Markov chains ergodic, even periodic ones.<a href="#cite_note-51">[51]</a> In fact, merely irreducible Markov chains correspond to <a href="/wiki/Ergodicity" title="Ergodicity">ergodic processes</a>, defined according to <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a>.<a href="#cite_note-:2-52">[52]</a> Some authors call a matrix primitive iff there exists some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> such that all entries of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd66ef6f4af0b81e278ea2a32de5d5d7d3ea6613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.587ex; height:2.676ex;" alt="{\displaystyle M^{k}}"> are positive.<a href="#cite_note-53">[53]</a> Some authors call it regular.<a href="#cite_note-54">[54]</a> <div class="mw-heading mw-heading4"><h4 id="Index_of_primitivity">Index of primitivity</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=26" title="Edit section: Index of primitivity">edit</a>]</div> The index of primitivity, or exponent, of a regular matrix, is the smallest <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> such that all entries of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd66ef6f4af0b81e278ea2a32de5d5d7d3ea6613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.587ex; height:2.676ex;" alt="{\displaystyle M^{k}}"> are positive. The exponent is purely a graph-theoretic property, since it depends only on whether each entry of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is zero or positive, and therefore can be found on a directed graph with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {sign} (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {sign} (M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d041ff01249465393f55c5d75634d74854731d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.27ex; height:2.843ex;" alt="{\displaystyle \mathrm {sign} (M)}"> as its adjacency matrix. There are several combinatorial results about the exponent when there are finitely many states. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> be the number of states, then<a href="#cite_note-55">[55]</a> <ul><li>The exponent is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq (n-1)^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq (n-1)^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ce628c5ede26bbbcb4ea0e81dc69b343f4aac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.717ex; height:3.176ex;" alt="{\displaystyle \leq (n-1)^{2}+1}">. The only case where it is an equality is when the graph of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> goes like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\to 2\to \dots \to n\to 1{\text{ and }}2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">→</mo> <mn>2</mn> <mo stretchy="false">→</mo> <mo>⋯</mo> <mo stretchy="false">→</mo> <mi>n</mi> <mo stretchy="false">→</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\to 2\to \dots \to n\to 1{\text{ and }}2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912ca6836de784a66b2337cb97ceb890587187e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:28.133ex; height:2.176ex;" alt="{\displaystyle 1\to 2\to \dots \to n\to 1{\text{ and }}2}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}"> diagonal entries, then its exponent is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq 2n-k-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq 2n-k-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d66356184a33a481d1c22377f15610601f0764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.065ex; height:2.343ex;" alt="{\displaystyle \leq 2n-k-1}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {sign} (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {sign} (M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d041ff01249465393f55c5d75634d74854731d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.27ex; height:2.843ex;" alt="{\displaystyle \mathrm {sign} (M)}"> is symmetric, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9c3e3aa1aec0e135b5d342181d9ee7f9814f87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.553ex; height:2.676ex;" alt="{\displaystyle M^{2}}"> has positive diagonal entries, which by previous proposition means its exponent is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq 2n-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq 2n-2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1548cafe6f7ca0a91ba733ecde279a20f07e2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.013ex; height:2.343ex;" alt="{\displaystyle \leq 2n-2}">.</li> <li>(Dulmage-Mendelsohn theorem) The exponent is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq n+s(n-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mi>n</mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq n+s(n-2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9590aa62527d932248a67afa7df4cfee83575e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.986ex; height:2.843ex;" alt="{\displaystyle \leq n+s(n-2)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> is the <a href="/wiki/Girth_(graph_theory)" title="Girth (graph theory)">girth of the graph</a>. It can be improved to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq (d+1)+s(d+1-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq (d+1)+s(d+1-2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0eb92afecc41b25eea3779530cbd09dd8415255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.443ex; height:2.843ex;" alt="{\displaystyle \leq (d+1)+s(d+1-2)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> is the <a href="/wiki/Diameter_(graph_theory)" title="Diameter (graph theory)">diameter of the graph</a>.<a href="#cite_note-56">[56]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Measure-preserving_dynamical_system">Measure-preserving dynamical system</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=27" title="Edit section: Measure-preserving dynamical system">edit</a>]</div> If a Markov chain has a stationary distribution, then it can be converted to a <a href="/wiki/Measure-preserving_dynamical_system" title="Measure-preserving dynamical system">measure-preserving dynamical system</a>: Let the probability space be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\Sigma ^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =\Sigma ^{\mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb540fb66c820cf8a052819eb29061ce739ca9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.873ex; height:2.676ex;" alt="{\displaystyle \Omega =\Sigma ^{\mathbb {N} }}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"> is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets. Let the probability measure be generated by the stationary distribution, and the Markov chain transition. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:\Omega \to \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:\Omega \to \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1aa99370a83771c64b1dada731520ce2237b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.544ex; height:2.176ex;" alt="{\displaystyle T:\Omega \to \Omega }"> be the shift operator: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X_{0},X_{1},\dots )=(X_{1},\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(X_{0},X_{1},\dots )=(X_{1},\dots )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecffb917a602978a77a26c1fdd9c0fe12a06d52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.837ex; height:2.843ex;" alt="{\displaystyle T(X_{0},X_{1},\dots )=(X_{1},\dots )}">. Similarly we can construct such a dynamical system with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\Sigma ^{\mathbb {Z} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =\Sigma ^{\mathbb {Z} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f9361be463ddb8b3a50715e919b726a09f96c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.783ex; height:2.676ex;" alt="{\displaystyle \Omega =\Sigma ^{\mathbb {Z} }}"> instead.<a href="#cite_note-57">[57]</a> Since irreducible Markov chains with finite state spaces have a unique stationary distribution, the above construction is unambiguous for irreducible Markov chains. In <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a>, a measure-preserving dynamical system is called "ergodic" iff any measurable subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{-1}(S)=S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{-1}(S)=S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f24b4e115b04bae02baf93591af676b82cd9fea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.959ex; height:3.176ex;" alt="{\displaystyle T^{-1}(S)=S}"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7c38be46a9b3fe15e5893edc24def5ba893e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.76ex; height:2.509ex;" alt="{\displaystyle S=\emptyset }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> (up to a null set). The terminology is inconsistent. Given a Markov chain with a stationary distribution that is strictly positive on all states, the Markov chain is irreducible iff its corresponding measure-preserving dynamical system is ergodic.<a href="#cite_note-:2-52">[52]</a> <div class="mw-heading mw-heading3"><h3 id="Markovian_representations">Markovian representations</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=28" title="Edit section: Markovian representations">edit</a>]</div> In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the "current" and "future" states. For example, let X be a non-Markovian process. Then define a process Y, such that each state of Y represents a time-interval of states of X. Mathematically, this takes the form: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t)={\big \{}X(s):s\in [a(t),b(t)]\,{\big \}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y(t)={\big \{}X(s):s\in [a(t),b(t)]\,{\big \}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d6e381d59b76a48ff453d6a16129ba7f2fd239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.866ex; height:3.176ex;" alt="{\displaystyle Y(t)={\big \{}X(s):s\in [a(t),b(t)]\,{\big \}}.}"></dd></dl> If Y has the Markov property, then it is a Markovian representation of X. An example of a non-Markovian process with a Markovian representation is an <a href="/wiki/Autoregressive_model" title="Autoregressive model">autoregressive</a> <a href="/wiki/Time_series" title="Time series">time series</a> of order greater than one.<a href="#cite_note-58">[58]</a> <div class="mw-heading mw-heading3"><h3 id="Hitting_times">Hitting times</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=29" title="Edit section: Hitting times">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type distribution</a></div>The hitting time is the time, starting in a given set of states until the chain arrives in a given state or set of states. The distribution of such a time period has a phase type distribution. The simplest such distribution is that of a single exponentially distributed transition. <div class="mw-heading mw-heading4"><h4 id="Expected_hitting_times">Expected hitting times</h4>[<a href="/w/index.php?title=Markov_chain&action=edit&section=30" title="Edit section: Expected hitting times">edit</a>]</div> For a subset of states A ⊆ S, the vector kA of hitting times (where element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}^{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f780d0aa6de52f1d926d13e5d00a886f3e50a6db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.676ex; height:3.176ex;" alt="{\displaystyle k_{i}^{A}}"> represents the <a href="/wiki/Expected_value" title="Expected value">expected value</a>, starting in state i that the chain enters one of the states in the set A) is the minimal non-negative solution to<a href="#cite_note-norris2-59">[59]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}k_{i}^{A}=0&{\text{ for }}i\in A\\-\sum _{j\in S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>i</mi> <mo>∈</mo> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>i</mi> <mo>∉</mo> <mi>A</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}k_{i}^{A}=0&{\text{ for }}i\in A\\-\sum _{j\in S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c21f1b0574a57f844a4889f486e6951117066b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.12ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}k_{i}^{A}=0&{\text{ for }}i\in A\\-\sum _{j\in S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Time_reversal">Time reversal</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=31" title="Edit section: Time reversal">edit</a>]</div> For a CTMC Xt, the time-reversed process is defined to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {X}}_{t}=X_{T-t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>−</mo> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}_{t}=X_{T-t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e9acc7406eed52ca23a7611682a4c3d7ad4f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.09ex; height:3.176ex;" alt="{\displaystyle {\hat {X}}_{t}=X_{T-t}}">. By <a href="/wiki/Kelly%27s_lemma" title="Kelly's lemma">Kelly's lemma</a> this process has the same stationary distribution as the forward process. A chain is said to be reversible if the reversed process is the same as the forward process. <a href="/wiki/Kolmogorov%27s_criterion" title="Kolmogorov's criterion">Kolmogorov's criterion</a> states that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions. <div class="mw-heading mw-heading3"><h3 id="Embedded_Markov_chain">Embedded Markov chain</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=32" title="Edit section: Embedded Markov chain">edit</a>]</div> One method of finding the <a href="/wiki/Stationary_probability_distribution" class="mw-redirect" title="Stationary probability distribution">stationary probability distribution</a>, π, of an <a href="/wiki/Ergodic" class="mw-redirect" title="Ergodic">ergodic</a> continuous-time Markov chain, Q, is by first finding its embedded Markov chain (EMC). Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a <a href="/wiki/Jump_process" title="Jump process">jump process</a>. Each element of the one-step transition probability matrix of the EMC, S, is denoted by sij, and represents the <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a> of transitioning from state i into state j. These conditional probabilities may be found by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{ij}={\begin{cases}{\frac {q_{ij}}{\sum _{k\neq i}q_{ik}}}&{\text{if }}i\neq j\\0&{\text{otherwise}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{ij}={\begin{cases}{\frac {q_{ij}}{\sum _{k\neq i}q_{ik}}}&{\text{if }}i\neq j\\0&{\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a471c811394ca548abeb1689144b16c2451dc96c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.207ex; height:7.509ex;" alt="{\displaystyle s_{ij}={\begin{cases}{\frac {q_{ij}}{\sum _{k\neq i}q_{ik}}}&{\text{if }}i\neq j\\0&{\text{otherwise}}.\end{cases}}}"></dd></dl> From this, S may be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=I-\left(\operatorname {diag} (Q)\right)^{-1}Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>I</mi> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=I-\left(\operatorname {diag} (Q)\right)^{-1}Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e7cea16ee57a5151e1eb680f834ae37730384c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.502ex; height:3.343ex;" alt="{\displaystyle S=I-\left(\operatorname {diag} (Q)\right)^{-1}Q}"></dd></dl> where I is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> and diag(Q) is the <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> formed by selecting the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> from the matrix Q and setting all other elements to zero. To find the stationary probability distribution vector, we must next find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi S=\varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mi>S</mi> <mo>=</mo> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi S=\varphi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb24ca860428d501034b5687d8274590e804b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.285ex; height:2.676ex;" alt="{\displaystyle \varphi S=\varphi ,}"></dd></dl> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> being a row vector, such that all elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> are greater than 0 and <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\varphi \|_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>φ</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\varphi \|_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0131469072b009c7d7d3d52c8a812e0e1fe537d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.899ex; height:2.843ex;" alt="{\displaystyle \|\varphi \|_{1}}"></a> = 1. From this, π may be found as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7ceda34e9f792545e52e5379830c37a4e61944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.867ex; height:6.676ex;" alt="{\displaystyle \pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.}"></dd></dl> (S may be periodic, even if Q is not. Once π is found, it must be normalized to a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a>.) Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton—the (discrete-time) Markov chain formed by observing X(t) at intervals of δ units of time. The random variables X(0), X(δ), X(2δ), ... give the sequence of states visited by the δ-skeleton. <div class="mw-heading mw-heading2"><h2 id="Special_types_of_Markov_chains">Special types of Markov chains</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=33" title="Edit section: Special types of Markov chains">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Markov_model">Markov model</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=34" title="Edit section: Markov model">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Markov_model" title="Markov model">Markov model</a></div> Markov models are used to model changing systems. There are 4 main types of models, that generalize Markov chains depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made: <table class="wikitable" style="border-spacing: 2px; border: 1px solid darkgray;"> <tbody><tr> <th> </th> <th>System state is fully observable </th> <th>System state is partially observable </th></tr> <tr> <th>System is autonomous </th> <td>Markov chain </td> <td><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model</a> </td></tr> <tr> <th>System is controlled </th> <td><a href="/wiki/Markov_decision_process" title="Markov decision process">Markov decision process</a> </td> <td><a href="/wiki/Partially_observable_Markov_decision_process" title="Partially observable Markov decision process">Partially observable Markov decision process</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Bernoulli_scheme">Bernoulli scheme</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=35" title="Edit section: Bernoulli scheme">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bernoulli_scheme" title="Bernoulli scheme">Bernoulli scheme</a></div> A <a href="/wiki/Bernoulli_scheme" title="Bernoulli scheme">Bernoulli scheme</a> is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is independent of even the current state (in addition to being independent of the past states). A Bernoulli scheme with only two possible states is known as a <a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a>. Note, however, by the <a href="/wiki/Ornstein_isomorphism_theorem" title="Ornstein isomorphism theorem">Ornstein isomorphism theorem</a>, that every aperiodic and irreducible Markov chain is isomorphic to a Bernoulli scheme;<a href="#cite_note-nicol-60">[60]</a> thus, one might equally claim that Markov chains are a "special case" of Bernoulli schemes. The isomorphism generally requires a complicated recoding. The isomorphism theorem is even a bit stronger: it states that any <a href="/wiki/Stationary_stochastic_process" class="mw-redirect" title="Stationary stochastic process">stationary stochastic process</a> is isomorphic to a Bernoulli scheme; the Markov chain is just one such example. <div class="mw-heading mw-heading3"><h3 id="Subshift_of_finite_type">Subshift of finite type</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=36" title="Edit section: Subshift of finite type">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Subshift_of_finite_type" title="Subshift of finite type">Subshift of finite type</a></div> When the Markov matrix is replaced by the <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of a <a href="/wiki/Finite_graph" class="mw-redirect" title="Finite graph">finite graph</a>, the resulting shift is termed a topological Markov chain or a subshift of finite type.<a href="#cite_note-nicol-60">[60]</a> A Markov matrix that is compatible with the adjacency matrix can then provide a <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a> on the subshift. Many chaotic <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a> are isomorphic to topological Markov chains; examples include <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a> of <a href="/wiki/Closed_manifold" title="Closed manifold">closed manifolds</a>, the <a href="/wiki/Thue%E2%80%93Morse_sequence" title="Thue–Morse sequence">Prouhet–Thue–Morse system</a>, the <a href="/w/index.php?title=Chacon_system&action=edit&redlink=1" class="new" title="Chacon system (page does not exist)">Chacon system</a>, <a href="/wiki/Sofic_system" class="mw-redirect" title="Sofic system">sofic systems</a>, <a href="/w/index.php?title=Context-free_system&action=edit&redlink=1" class="new" title="Context-free system (page does not exist)">context-free systems</a> and <a href="/w/index.php?title=Block-coding_system&action=edit&redlink=1" class="new" title="Block-coding system (page does not exist)">block-coding systems</a>.<a href="#cite_note-nicol-60">[60]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=37" title="Edit section: Applications">edit</a>]</div> Markov chains have been employed in a wide range of topics across the natural and social sciences, and in technological applications. They have been used for forecasting in several areas: for example, price trends,<a href="#cite_note-SLS-61">[61]</a> wind power,<a href="#cite_note-CGLT-62">[62]</a> <a href="/wiki/Stochastic_terrorism" title="Stochastic terrorism">stochastic terrorism</a>,<a href="#cite_note-Woo2002-63">[63]</a><a href="#cite_note-Woo2003-64">[64]</a> and <a href="/wiki/Solar_irradiance" title="Solar irradiance">solar irradiance</a>.<a href="#cite_note-MMW-65">[65]</a> The Markov chain forecasting models utilize a variety of settings, from discretizing the time series,<a href="#cite_note-CGLT-62">[62]</a> to hidden Markov models combined with wavelets,<a href="#cite_note-SLS-61">[61]</a> and the Markov chain mixture distribution model (MCM).<a href="#cite_note-MMW-65">[65]</a> <div class="mw-heading mw-heading3"><h3 id="Physics">Physics</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=38" title="Edit section: Physics">edit</a>]</div> Markovian systems appear extensively in <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> and <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.<a href="#cite_note-66">[66]</a><a href="#cite_note-auto1-67">[67]</a> For example, a thermodynamic state operates under a probability distribution that is difficult or expensive to acquire. Therefore, Markov Chain Monte Carlo method can be used to draw samples randomly from a black-box to approximate the probability distribution of attributes over a range of objects.<a href="#cite_note-auto1-67">[67]</a> Markov chains are used in <a href="/wiki/Lattice_QCD" title="Lattice QCD">lattice QCD</a> simulations.<a href="#cite_note-68">[68]</a> <div class="mw-heading mw-heading3"><h3 id="Chemistry">Chemistry</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=39" title="Edit section: Chemistry">edit</a>]</div> <div class="thumb tleft" style=""><div class="thumbinner" style="width:202px"><div class="thumbimage noresize" style="width:200px;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {{E}+{\underset {Substrate \atop binding}{S<=>E}}{\overset {Catalytic \atop step}{S->E}}+P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>E</mtext> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mtext>S</mtext> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↽</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">⇀</mo> </mrow> </mrow> </mstyle> </mrow> </mover> </mrow> <mtext>E</mtext> </mrow> <mfrac linethickness="0"> <mtext>Substrate</mtext> <mtext>binding</mtext> </mfrac> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mtext>S</mtext> <mo stretchy="false">⟶</mo> <mtext>E</mtext> </mrow> <mfrac linethickness="0"> <mtext>Catalytic</mtext> <mtext>step</mtext> </mfrac> </mover> </mrow> <mo>+</mo> <mtext>P</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {{E}+{\underset {Substrate \atop binding}{S<=>E}}{\overset {Catalytic \atop step}{S->E}}+P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d1ad710a84b5451e91bf67b1bdac85e4c738fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.341ex; height:8.676ex;" alt="{\displaystyle {\ce {{E}+{\underset {Substrate \atop binding}{S<=>E}}{\overset {Catalytic \atop step}{S->E}}+P}}}"></div><div class="thumbcaption"><a href="/wiki/Michaelis-Menten_kinetics" class="mw-redirect" title="Michaelis-Menten kinetics">Michaelis-Menten kinetics</a>. The enzyme (E) binds a substrate (S) and produces a product (P). Each reaction is a state transition in a Markov chain.</div></div></div>A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain.<a href="#cite_note-69">[69]</a> Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property. For example, imagine a large number n of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate. Perhaps the molecule is an enzyme, and the states refer to how it is folded. The state of any single enzyme follows a Markov chain, and since the molecules are essentially independent of each other, the number of molecules in state A or B at a time is n times the probability a given molecule is in that state. The classical model of enzyme activity, <a href="/wiki/Michaelis%E2%80%93Menten_kinetics" title="Michaelis–Menten kinetics">Michaelis–Menten kinetics</a>, can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction. While Michaelis-Menten is fairly straightforward, far more complicated reaction networks can also be modeled with Markov chains.<a href="#cite_note-70">[70]</a> An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicals <a href="/wiki/In_silico" title="In silico">in silico</a> towards a desired class of compounds such as drugs or natural products.<a href="#cite_note-71">[71]</a> As a molecule is grown, a fragment is selected from the nascent molecule as the "current" state. It is not aware of its past (that is, it is not aware of what is already bonded to it). It then transitions to the next state when a fragment is attached to it. The transition probabilities are trained on databases of authentic classes of compounds.<a href="#cite_note-72">[72]</a> Also, the growth (and composition) of <a href="/wiki/Copolymer" title="Copolymer">copolymers</a> may be modeled using Markov chains. Based on the reactivity ratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated (for example, whether monomers tend to add in alternating fashion or in long runs of the same monomer). Due to <a href="/wiki/Steric_effects" title="Steric effects">steric effects</a>, second-order Markov effects may also play a role in the growth of some polymer chains. Similarly, it has been suggested that the crystallization and growth of some epitaxial <a href="/wiki/Superlattice" title="Superlattice">superlattice</a> oxide materials can be accurately described by Markov chains.<a href="#cite_note-73">[73]</a> <div class="mw-heading mw-heading3"><h3 id="Biology">Biology</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=40" title="Edit section: Biology">edit</a>]</div> Markov chains are used in various areas of biology. Notable examples include: <ul><li><a href="/wiki/Phylogenetics" title="Phylogenetics">Phylogenetics</a> and <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>, where most <a href="/wiki/Models_of_DNA_evolution" title="Models of DNA evolution">models of DNA evolution</a> use continuous-time Markov chains to describe the <a href="/wiki/Nucleotide" title="Nucleotide">nucleotide</a> present at a given site in the <a href="/wiki/Genome" title="Genome">genome</a>.</li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a>, where Markov chains are in particular a central tool in the theoretical study of <a href="/wiki/Matrix_population_models" title="Matrix population models">matrix population models</a>.</li> <li><a href="/wiki/Neurobiology" class="mw-redirect" title="Neurobiology">Neurobiology</a>, where Markov chains have been used, e.g., to simulate the mammalian neocortex.<a href="#cite_note-74">[74]</a></li> <li><a href="/wiki/Systems_biology" title="Systems biology">Systems biology</a>, for instance with the modeling of viral infection of single cells.<a href="#cite_note-75">[75]</a></li> <li><a href="/wiki/Compartmental_models_in_epidemiology" title="Compartmental models in epidemiology">Compartmental models</a> for disease outbreak and epidemic modeling.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Testing">Testing</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=41" title="Edit section: Testing">edit</a>]</div> Several theorists have proposed the idea of the Markov chain statistical test (MCST), a method of conjoining Markov chains to form a "<a href="/wiki/Markov_blanket" title="Markov blanket">Markov blanket</a>", arranging these chains in several recursive layers ("wafering") and producing more efficient test sets—samples—as a replacement for exhaustive testing.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Solar_irradiance_variability">Solar irradiance variability</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=42" title="Edit section: Solar irradiance variability">edit</a>]</div> <a href="/wiki/Solar_irradiance" title="Solar irradiance">Solar irradiance</a> variability assessments are useful for <a href="/wiki/Solar_power" title="Solar power">solar power</a> applications. Solar irradiance variability at any location over time is mainly a consequence of the deterministic variability of the sun's path across the sky dome and the variability in cloudiness. The variability of accessible solar irradiance on Earth's surface has been modeled using Markov chains,<a href="#cite_note-76">[76]</a><a href="#cite_note-77">[77]</a><a href="#cite_note-78">[78]</a><a href="#cite_note-79">[79]</a> also including modeling the two states of clear and cloudiness as a two-state Markov chain.<a href="#cite_note-80">[80]</a><a href="#cite_note-81">[81]</a> <div class="mw-heading mw-heading3"><h3 id="Speech_recognition">Speech recognition</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=43" title="Edit section: Speech recognition">edit</a>]</div> <a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov models</a> have been used in <a href="/wiki/Speech_recognition#Hidden_Markov_models" title="Speech recognition">automatic speech recognition</a> systems.<a href="#cite_note-82">[82]</a> <div class="mw-heading mw-heading3"><h3 id="Information_theory">Information theory</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=44" title="Edit section: Information theory">edit</a>]</div> Markov chains are used throughout information processing. <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>'s famous 1948 paper <a href="/wiki/A_Mathematical_Theory_of_Communication" title="A Mathematical Theory of Communication">A Mathematical Theory of Communication</a>, which in a single step created the field of <a href="/wiki/Information_theory" title="Information theory">information theory</a>, opens by introducing the concept of <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">entropy</a> by modeling texts in a natural language (such as English) as generated by an ergodic Markov process, where each letter may depend statistically on previous letters.<a href="#cite_note-83">[83]</a> Such idealized models can capture many of the statistical regularities of systems. Even without describing the full structure of the system perfectly, such signal models can make possible very effective <a href="/wiki/Data_compression" title="Data compression">data compression</a> through <a href="/wiki/Entropy_encoding" class="mw-redirect" title="Entropy encoding">entropy encoding</a> techniques such as <a href="/wiki/Arithmetic_coding" title="Arithmetic coding">arithmetic coding</a>. They also allow effective <a href="/wiki/State_estimation" class="mw-redirect" title="State estimation">state estimation</a> and <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a>. Markov chains also play an important role in <a href="/wiki/Reinforcement_learning" title="Reinforcement learning">reinforcement learning</a>. Markov chains are also the basis for hidden Markov models, which are an important tool in such diverse fields as telephone networks (which use the <a href="/wiki/Viterbi_algorithm" title="Viterbi algorithm">Viterbi algorithm</a> for error correction), speech recognition and <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a> (such as in rearrangements detection<a href="#cite_note-rearrang-84">[84]</a>). The <a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Markov_chain_algorithm" title="Lempel–Ziv–Markov chain algorithm">LZMA</a> lossless data compression algorithm combines Markov chains with <a href="/wiki/LZ77_and_LZ78" title="LZ77 and LZ78">Lempel-Ziv compression</a> to achieve very high compression ratios. <div class="mw-heading mw-heading3"><h3 id="Queueing_theory">Queueing theory</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=45" title="Edit section: Queueing theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></div>Markov chains are the basis for the analytical treatment of queues (<a href="/wiki/Queueing_theory" title="Queueing theory">queueing theory</a>). <a href="/wiki/Agner_Krarup_Erlang" title="Agner Krarup Erlang">Agner Krarup Erlang</a> initiated the subject in 1917.<a href="#cite_note-MacTutor|id=Erlang-85">[85]</a> This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).<a href="#cite_note-CTCN-86">[86]</a> Numerous queueing models use continuous-time Markov chains. For example, an <a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1 queue</a> is a CTMC on the non-negative integers where upward transitions from i to i + 1 occur at rate λ according to a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> and describe job arrivals, while transitions from i to i – 1 (for i > 1) occur at rate μ (job service times are exponentially distributed) and describe completed services (departures) from the queue. <div class="mw-heading mw-heading3"><h3 id="Internet_applications">Internet applications</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=46" title="Edit section: Internet applications">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PageRank_with_Markov_Chain.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/PageRank_with_Markov_Chain.png/220px-PageRank_with_Markov_Chain.png" decoding="async" width="220" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/PageRank_with_Markov_Chain.png/330px-PageRank_with_Markov_Chain.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/PageRank_with_Markov_Chain.png/440px-PageRank_with_Markov_Chain.png 2x" data-file-width="1080" data-file-height="776" /></a><figcaption>A state diagram that represents the PageRank algorithm with a transitional probability of M, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c993bcf539084011d0805050262d33b8ee5670ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.014ex; height:5.676ex;" alt="{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}">.</figcaption></figure> The <a href="/wiki/PageRank" title="PageRank">PageRank</a> of a webpage as used by <a href="/wiki/Google" title="Google">Google</a> is defined by a Markov chain.<a href="#cite_note-87">[87]</a><a href="#cite_note-BrijP.2016-88">[88]</a><a href="#cite_note-LangvilleMeyer2006-89">[89]</a> It is the probability to be at page <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> in the stationary distribution on the following Markov chain on all (known) webpages. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> is the number of known webpages, and a page <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"> links to it then it has transition probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c993bcf539084011d0805050262d33b8ee5670ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.014ex; height:5.676ex;" alt="{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}"> for all pages that are linked to and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-\alpha }{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-\alpha }{N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd3f178a8563113a155e40c4a3c04bead8ea280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.327ex; height:5.176ex;" alt="{\displaystyle {\frac {1-\alpha }{N}}}"> for all pages that are not linked to. The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> is taken to be about 0.15.<a href="#cite_note-pagerank-90">[90]</a> Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an individual user.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Statistics">Statistics</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=47" title="Edit section: Statistics">edit</a>]</div> Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> (MCMC). In recent years this has revolutionized the practicability of <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a> methods, allowing a wide range of <a href="/wiki/Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior distributions</a> to be simulated and their parameters found numerically.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Conflict_and_combat">Conflict and combat</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=48" title="Edit section: Conflict and combat">edit</a>]</div> In 1971 a <a href="/wiki/Naval_Postgraduate_School" title="Naval Postgraduate School">Naval Postgraduate School</a> Master's thesis proposed to model a variety of combat between adversaries as a Markov chain "with states reflecting the control, maneuver, target acquisition, and target destruction actions of a weapons system" and discussed the parallels between the resulting Markov chain and <a href="/wiki/Lanchester%27s_laws" title="Lanchester's laws">Lanchester's laws</a>.<a href="#cite_note-dtic1-91">[91]</a> In 1975 Duncan and Siverson remarked that Markov chains could be used to model conflict between state actors, and thought that their analysis would help understand "the behavior of social and political organizations in situations of conflict."<a href="#cite_note-duncan75-92">[92]</a> <div class="mw-heading mw-heading3"><h3 id="Economics_and_finance">Economics and finance</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=49" title="Edit section: Economics and finance">edit</a>]</div> Markov chains are used in finance and economics to model a variety of different phenomena, including the distribution of income, the size distribution of firms, asset prices and market crashes. <a href="/wiki/D._G._Champernowne" title="D. G. Champernowne">D. G. Champernowne</a> built a Markov chain model of the distribution of income in 1953.<a href="#cite_note-93">[93]</a> <a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert A. Simon</a> and co-author Charles Bonini used a Markov chain model to derive a stationary Yule distribution of firm sizes.<a href="#cite_note-94">[94]</a> <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a> was the first to observe that stock prices followed a random walk.<a href="#cite_note-95">[95]</a> The random walk was later seen as evidence in favor of the <a href="/wiki/Efficient-market_hypothesis" title="Efficient-market hypothesis">efficient-market hypothesis</a> and random walk models were popular in the literature of the 1960s.<a href="#cite_note-96">[96]</a> Regime-switching models of business cycles were popularized by <a href="/wiki/James_D._Hamilton" title="James D. Hamilton">James D. Hamilton</a> (1989), who used a Markov chain to model switches between periods of high and low GDP growth (or, alternatively, economic expansions and recessions).<a href="#cite_note-97">[97]</a> A more recent example is the <a href="/wiki/Markov_switching_multifractal" title="Markov switching multifractal">Markov switching multifractal</a> model of <a href="/wiki/Laurent_E._Calvet" class="mw-redirect" title="Laurent E. Calvet">Laurent E. Calvet</a> and Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models.<a href="#cite_note-98">[98]</a><a href="#cite_note-99">[99]</a> It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns. Dynamic macroeconomics makes heavy use of Markov chains. An example is using Markov chains to exogenously model prices of equity (stock) in a <a href="/wiki/General_equilibrium" class="mw-redirect" title="General equilibrium">general equilibrium</a> setting.<a href="#cite_note-100">[100]</a> <a href="/wiki/Credit_rating_agency" title="Credit rating agency">Credit rating agencies</a> produce annual tables of the transition probabilities for bonds of different credit ratings.<a href="#cite_note-101">[101]</a> <div class="mw-heading mw-heading3"><h3 id="Social_sciences">Social sciences</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=50" title="Edit section: Social sciences">edit</a>]</div> Markov chains are generally used in describing <a href="/wiki/Path-dependent" class="mw-redirect" title="Path-dependent">path-dependent</a> arguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due to <a href="/wiki/Karl_Marx" title="Karl Marx">Karl Marx</a>'s <a href="/wiki/Das_Kapital" title="Das Kapital">Das Kapital</a>, tying <a href="/wiki/Economic_development" title="Economic development">economic development</a> to the rise of <a href="/wiki/Capitalism" title="Capitalism">capitalism</a>. In current research, it is common to use a Markov chain to model how once a country reaches a specific level of economic development, the configuration of structural factors, such as size of the <a href="/wiki/Middle_class" title="Middle class">middle class</a>, the ratio of urban to rural residence, the rate of <a href="/wiki/Political" class="mw-redirect" title="Political">political</a> mobilization, etc., will generate a higher probability of transitioning from <a href="/wiki/Authoritarian" class="mw-redirect" title="Authoritarian">authoritarian</a> to <a href="/wiki/Democratic_regime" class="mw-redirect" title="Democratic regime">democratic regime</a>.<a href="#cite_note-102">[102]</a> <div class="mw-heading mw-heading3"><h3 id="Music">Music</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=51" title="Edit section: Music">edit</a>]</div> Markov chains are employed in <a href="/wiki/Algorithmic_composition" title="Algorithmic composition">algorithmic music composition</a>, particularly in <a href="/wiki/Software" title="Software">software</a> such as <a href="/wiki/Csound" title="Csound">Csound</a>, <a href="/wiki/Max_(software)" title="Max (software)">Max</a>, and <a href="/wiki/SuperCollider" title="SuperCollider">SuperCollider</a>. In a first-order chain, the states of the system become note or pitch values, and a <a href="/wiki/Probability_vector" title="Probability vector">probability vector</a> for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce output note values based on the transition matrix weightings, which could be <a href="/wiki/MIDI" title="MIDI">MIDI</a> note values, frequency (<a href="/wiki/Hertz" title="Hertz">Hz</a>), or any other desirable metric.<a href="#cite_note-103">[103]</a> <table class="wikitable" style="float: left"> <caption>1st-order matrix </caption> <tbody><tr> <th>Note</th> <th>A</th> <th>C♯</th> <th>E♭ </th></tr> <tr> <th>A </th> <td>0.1</td> <td>0.6</td> <td>0.3 </td></tr> <tr> <th>C♯ </th> <td>0.25</td> <td>0.05</td> <td>0.7 </td></tr> <tr> <th>E♭ </th> <td>0.7</td> <td>0.3</td> <td>0 </td></tr></tbody></table> <table class="wikitable" style="float: left; margin-left: 1em"> <caption>2nd-order matrix </caption> <tbody><tr> <th>Notes</th> <th>A</th> <th>D</th> <th>G </th></tr> <tr> <th>AA </th> <td>0.18</td> <td>0.6</td> <td>0.22 </td></tr> <tr> <th>AD </th> <td>0.5</td> <td>0.5</td> <td>0 </td></tr> <tr> <th>AG </th> <td>0.15</td> <td>0.75</td> <td>0.1 </td></tr> <tr> <th>DD </th> <td>0</td> <td>0</td> <td>1 </td></tr> <tr> <th>DA </th> <td>0.25</td> <td>0</td> <td>0.75 </td></tr> <tr> <th>DG </th> <td>0.9</td> <td>0.1</td> <td>0 </td></tr> <tr> <th>GG </th> <td>0.4</td> <td>0.4</td> <td>0.2 </td></tr> <tr> <th>GA </th> <td>0.5</td> <td>0.25</td> <td>0.25 </td></tr> <tr> <th>GD </th> <td>1</td> <td>0</td> <td>0 </td></tr></tbody></table> <div style="clear:both;" class=""></div> A second-order Markov chain can be introduced by considering the current state and also the previous state, as indicated in the second table. Higher, nth-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of <a href="/wiki/Phrase_(music)" title="Phrase (music)">phrasal</a> structure, rather than the 'aimless wandering' produced by a first-order system.<a href="#cite_note-Roads-104">[104]</a> Markov chains can be used structurally, as in Xenakis's Analogique A and B.<a href="#cite_note-105">[105]</a> Markov chains are also used in systems which use a Markov model to react interactively to music input.<a href="#cite_note-106">[106]</a> Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory. In order to overcome this limitation, a new approach has been proposed.<a href="#cite_note-107">[107]</a> <div class="mw-heading mw-heading3"><h3 id="Games_and_sports">Games and sports</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=52" title="Edit section: Games and sports">edit</a>]</div> Markov chains can be used to model many games of chance. The children's games <a href="/wiki/Snakes_and_Ladders" class="mw-redirect" title="Snakes and Ladders">Snakes and Ladders</a> and "<a href="/wiki/Hi_Ho!_Cherry-O" title="Hi Ho! Cherry-O">Hi Ho! Cherry-O</a>", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (on a given square) and from there has fixed odds of moving to certain other states (squares).[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Markov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs and position of the runners. Mark Pankin shows that Markov chain models can be used to evaluate runs created for both individual players as well as a team.<a href="#cite_note-108">[108]</a> He also discusses various kinds of strategies and play conditions: how Markov chain models have been used to analyze statistics for game situations such as <a href="/wiki/Bunt_(baseball)" title="Bunt (baseball)">bunting</a> and <a href="/wiki/Base_stealing" class="mw-redirect" title="Base stealing">base stealing</a> and differences when playing on grass vs. <a href="/wiki/AstroTurf" title="AstroTurf">AstroTurf</a>.<a href="#cite_note-109">[109]</a> <div class="mw-heading mw-heading3"><h3 id="Markov_text_generators">Markov text generators</h3>[<a href="/w/index.php?title=Markov_chain&action=edit&section=53" title="Edit section: Markov text generators">edit</a>]</div> Markov processes can also be used to <a href="/wiki/Natural_language_generation" title="Natural language generation">generate superficially real-looking text</a> given a sample document. Markov processes are used in a variety of recreational "<a href="/wiki/Parody_generator" title="Parody generator">parody generator</a>" software (see <a href="/wiki/Dissociated_press" title="Dissociated press">dissociated press</a>, Jeff Harrison,<a href="#cite_note-110">[110]</a> <a href="/wiki/Mark_V._Shaney" title="Mark V. Shaney">Mark V. Shaney</a>,<a href="#cite_note-Travesty-111">[111]</a><a href="#cite_note-Hartman-112">[112]</a> and Academias Neutronium). Several open-source text generation libraries using Markov chains exist. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=54" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Dynamics_of_Markovian_particles" title="Dynamics of Markovian particles">Dynamics of Markovian particles</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov process</a></li> <li><a href="/wiki/Markov_chain_approximation_method" title="Markov chain approximation method">Markov chain approximation method</a></li> <li><a href="/wiki/Markov_chain_geostatistics" title="Markov chain geostatistics">Markov chain geostatistics</a></li> <li><a href="/wiki/Markov_chain_mixing_time" title="Markov chain mixing time">Markov chain mixing time</a></li> <li><a href="/wiki/Markov_chain_tree_theorem" title="Markov chain tree theorem">Markov chain tree theorem</a></li> <li><a href="/wiki/Markov_decision_process" title="Markov decision process">Markov decision process</a></li> <li><a href="/wiki/Markov_information_source" title="Markov information source">Markov information source</a></li> <li><a href="/wiki/Markov_odometer" title="Markov odometer">Markov odometer</a></li> <li><a href="/wiki/Markov_operator" title="Markov operator">Markov operator</a></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Master_equation" title="Master equation">Master equation</a></li> <li><a href="/wiki/Quantum_Markov_chain" title="Quantum Markov chain">Quantum Markov chain</a></li> <li><a href="/wiki/Semi-Markov_process" class="mw-redirect" title="Semi-Markov process">Semi-Markov process</a></li> <li><a href="/wiki/Stochastic_cellular_automaton" title="Stochastic cellular automaton">Stochastic cellular automaton</a></li> <li><a href="/wiki/Telescoping_Markov_chain" title="Telescoping Markov chain">Telescoping Markov chain</a></li> <li><a href="/wiki/Variable-order_Markov_model" title="Variable-order Markov model">Variable-order Markov model</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=55" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-MeynTweedie2009page3-1">^ <a href="#cite_ref-MeynTweedie2009page3_1-0">a</a> <a href="#cite_ref-MeynTweedie2009page3_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSean_MeynRichard_L._Tweedie2009" class="citation book cs1">Sean Meyn; Richard L. Tweedie (2 April 2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Md7RnYEPkJwC">Markov Chains and Stochastic Stability</a>. Cambridge University Press. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-73182-9" title="Special:BookSources/978-0-521-73182-9"><bdi>978-0-521-73182-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Markov+Chains+and+Stochastic+Stability&rft.pages=3&rft.pub=Cambridge+University+Press&rft.date=2009-04-02&rft.isbn=978-0-521-73182-9&rft.au=Sean+Meyn&rft.au=Richard+L.+Tweedie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMd7RnYEPkJwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-RubinsteinKroese2011page225-2"><a href="#cite_ref-RubinsteinKroese2011page225_2-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReuven_Y._RubinsteinDirk_P._Kroese2011" class="citation book cs1">Reuven Y. Rubinstein; Dirk P. 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John Wiley & Sons. p. 225. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-21052-9" title="Special:BookSources/978-1-118-21052-9"><bdi>978-1-118-21052-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Simulation+and+the+Monte+Carlo+Method&rft.pages=225&rft.pub=John+Wiley+%26+Sons&rft.date=2011-09-20&rft.isbn=978-1-118-21052-9&rft.au=Reuven+Y.+Rubinstein&rft.au=Dirk+P.+Kroese&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyWcvT80gQK4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-GamermanLopes2006-3"><a href="#cite_ref-GamermanLopes2006_3-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDani_GamermanHedibert_F._Lopes2006" class="citation book cs1">Dani Gamerman; Hedibert F. 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CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-587-0" title="Special:BookSources/978-1-58488-587-0"><bdi>978-1-58488-587-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Markov+Chain+Monte+Carlo%3A+Stochastic+Simulation+for+Bayesian+Inference%2C+Second+Edition&rft.pub=CRC+Press&rft.date=2006-05-10&rft.isbn=978-1-58488-587-0&rft.au=Dani+Gamerman&rft.au=Hedibert+F.+Lopes&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyPvECi_L3bwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-OxfordMarkovian-4"><a href="#cite_ref-OxfordMarkovian_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReference-OED-Markovian" class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.oed.com/search/dictionary/?q=Markovian">"Markovian"</a>. <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a> (Online ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Markovian&rft.btitle=Oxford+English+Dictionary&rft.edition=Online&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Fwww.oed.com%2Fsearch%2Fdictionary%2F%3Fq%3DMarkovian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> (Subscription or <a rel="nofollow" class="external text" href="https://www.oed.com/public/login/loggingin#withyourlibrary">participating institution membership</a> required.) </li> <li id="cite_note-:3-5"><a href="#cite_ref-:3_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFØksendal,_B._K._(Bernt_Karsten)2003" class="citation book cs1">Øksendal, B. 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Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3540047581" title="Special:BookSources/3540047581"><bdi>3540047581</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/52203046">52203046</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+differential+equations+%3A+an+introduction+with+applications&rft.place=Berlin&rft.edition=6th&rft.pub=Springer&rft.date=2003&rft_id=info%3Aoclcnum%2F52203046&rft.isbn=3540047581&rft.au=%C3%98ksendal%2C+B.+K.+%28Bernt+Karsten%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Asmussen2003page73-6">^ <a href="#cite_ref-Asmussen2003page73_6-0">a</a> <a href="#cite_ref-Asmussen2003page73_6-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSøren_Asmussen2003" class="citation book cs1">Søren Asmussen (15 May 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BeYaTxesKy0C">Applied Probability and Queues</a>. Springer Science & Business Media. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-00211-8" title="Special:BookSources/978-0-387-00211-8"><bdi>978-0-387-00211-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Probability+and+Queues&rft.pages=7&rft.pub=Springer+Science+%26+Business+Media&rft.date=2003-05-15&rft.isbn=978-0-387-00211-8&rft.au=S%C3%B8ren+Asmussen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBeYaTxesKy0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Parzen1999page1882-7"><a href="#cite_ref-Parzen1999page1882_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmanuel_Parzen2015" class="citation book cs1">Emanuel Parzen (17 June 2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0mB2CQAAQBAJ">Stochastic Processes</a>. Courier Dover Publications. p. 188. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-79688-8" title="Special:BookSources/978-0-486-79688-8"><bdi>978-0-486-79688-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Processes&rft.pages=188&rft.pub=Courier+Dover+Publications&rft.date=2015-06-17&rft.isbn=978-0-486-79688-8&rft.au=Emanuel+Parzen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0mB2CQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-KarlinTaylor2012page292-8"><a href="#cite_ref-KarlinTaylor2012page292_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamuel_KarlinHoward_E._Taylor2012" class="citation book cs1">Samuel Karlin; Howard E. Taylor (2 December 2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dSDxjX9nmmMC">A First Course in Stochastic Processes</a>. Academic Press. pp. 29 and 30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-057041-9" title="Special:BookSources/978-0-08-057041-9"><bdi>978-0-08-057041-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Stochastic+Processes&rft.pages=29+and+30&rft.pub=Academic+Press&rft.date=2012-12-02&rft.isbn=978-0-08-057041-9&rft.au=Samuel+Karlin&rft.au=Howard+E.+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdSDxjX9nmmMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Lamperti1977chap62-9"><a href="#cite_ref-Lamperti1977chap62_9-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Lamperti1977" class="citation book cs1">John Lamperti (1977). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Pd4cvgAACAAJ">Stochastic processes: a survey of the mathematical theory</a>. Springer-Verlag. pp. 106–121. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-90275-1" title="Special:BookSources/978-3-540-90275-1"><bdi>978-3-540-90275-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+processes%3A+a+survey+of+the+mathematical+theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E106-%3C%2Fspan%3E121&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=978-3-540-90275-1&rft.au=John+Lamperti&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPd4cvgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Ross1996page174and2312-10"><a href="#cite_ref-Ross1996page174and2312_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSheldon_M._Ross1996" class="citation book cs1">Sheldon M. 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CUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-81099-X" title="Special:BookSources/0-521-81099-X">0-521-81099-X</a> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Parzen, E. (1962) Stochastic Processes, Holden-Day. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8162-6664-6" title="Special:BookSources/0-8162-6664-6">0-8162-6664-6</a> (Table 6.1) </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-920613-9" title="Special:BookSources/0-19-920613-9">0-19-920613-9</a> (entry for "Markov chain") </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Dodge, Y. The Oxford Dictionary of Statistical Terms, OUP. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-920613-9" title="Special:BookSources/0-19-920613-9">0-19-920613-9</a> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Meyn, S. Sean P., and Richard L. Tweedie. (2009) Markov chains and stochastic stability. Cambridge University Press. (Preface, p. iii) </li> <li id="cite_note-GrinsteadSnell1997page4643-16">^ <a href="#cite_ref-GrinsteadSnell1997page4643_16-0">a</a> <a href="#cite_ref-GrinsteadSnell1997page4643_16-1">b</a> <a href="#cite_ref-GrinsteadSnell1997page4643_16-2">c</a> <a href="#cite_ref-GrinsteadSnell1997page4643_16-3">d</a> <a href="#cite_ref-GrinsteadSnell1997page4643_16-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCharles_Miller_GrinsteadJames_Laurie_Snell1997" class="citation book cs1">Charles Miller Grinstead; James Laurie Snell (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/flooved3489">Introduction to Probability</a>. 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Springer Science & Business Media. p. ix. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-3124-8" title="Special:BookSources/978-1-4757-3124-8"><bdi>978-1-4757-3124-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Markov+Chains%3A+Gibbs+Fields%2C+Monte+Carlo+Simulation%2C+and+Queues&rft.pages=ix&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013-03-09&rft.isbn=978-1-4757-3124-8&rft.au=Pierre+Bremaud&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjrPVBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Hayes20133-18">^ <a href="#cite_ref-Hayes20133_18-0">a</a> <a href="#cite_ref-Hayes20133_18-1">b</a> <a href="#cite_ref-Hayes20133_18-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayes2013" class="citation journal cs1">Hayes, Brian (2013). 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American Scientist. 101 (2): 92–96. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1511%2F2013.101.92">10.1511/2013.101.92</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Scientist&rft.atitle=First+links+in+the+Markov+chain&rft.volume=101&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E92-%3C%2Fspan%3E96&rft.date=2013&rft_id=info%3Adoi%2F10.1511%2F2013.101.92&rft.aulast=Hayes&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Ross1996page235and3583-19">^ <a href="#cite_ref-Ross1996page235and3583_19-0">a</a> <a href="#cite_ref-Ross1996page235and3583_19-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSheldon_M._Ross1996" class="citation book cs1">Sheldon M. 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Wiley. pp. 235 and 358. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-12062-9" title="Special:BookSources/978-0-471-12062-9"><bdi>978-0-471-12062-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+processes&rft.pages=235+and+358&rft.pub=Wiley&rft.date=1996&rft.isbn=978-0-471-12062-9&rft.au=Sheldon+M.+Ross&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DImUPAQAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-JarrowProtter20042-20"><a href="#cite_ref-JarrowProtter20042_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJarrowProtter2004" class="citation book cs1">Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: The early years, 1880–1970". 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MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-18158-7" title="Special:BookSources/978-0-262-18158-7"><bdi>978-0-262-18158-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Computer+Music+Tutorial&rft.pub=MIT+Press&rft.date=1996&rft.isbn=978-0-262-18158-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> Xenakis, Iannis; Kanach, Sharon (1992) Formalized Music: Mathematics and Thought in Composition, Pendragon Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1576470792" title="Special:BookSources/1576470792">1576470792</a> </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120713235933/http://www.csl.sony.fr/~pachet/">"Continuator"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.csl.sony.fr/~pachet/">the original</a> on July 13, 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Continuator&rft_id=http%3A%2F%2Fwww.csl.sony.fr%2F~pachet%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> Pachet, F.; Roy, P.; Barbieri, G. (2011) <a rel="nofollow" class="external text" href="http://www.csl.sony.fr/downloads/papers/2011/pachet-11b.pdf">"Finite-Length Markov Processes with Constraints"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120414183247/http://www.csl.sony.fr/downloads/papers/2011/pachet-11b.pdf">Archived</a> 2012-04-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI, pages 635–642, Barcelona, Spain, July 2011 </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPankin" class="citation web cs1">Pankin, Mark D. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071209122054/http://www.pankin.com/markov/theory.htm">"MARKOV CHAIN MODELS: THEORETICAL BACKGROUND"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.pankin.com/markov/theory.htm">the original</a> on 2007-12-09. Retrieved 2007-11-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=MARKOV+CHAIN+MODELS%3A+THEORETICAL+BACKGROUND&rft.aulast=Pankin&rft.aufirst=Mark+D.&rft_id=http%3A%2F%2Fwww.pankin.com%2Fmarkov%2Ftheory.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPankin" class="citation web cs1">Pankin, Mark D. <a rel="nofollow" class="external text" href="http://www.pankin.com/markov/intro.htm">"BASEBALL AS A MARKOV CHAIN"</a>. Retrieved 2009-04-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=BASEBALL+AS+A+MARKOV+CHAIN&rft.aulast=Pankin&rft.aufirst=Mark+D.&rft_id=http%3A%2F%2Fwww.pankin.com%2Fmarkov%2Fintro.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20101206043430/http://www.fieralingue.it/modules.php?name=Content&pa=list_pages_categories&cid=111">"Poet's Corner – Fieralingue"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.fieralingue.it/modules.php?name=Content&pa=list_pages_categories&cid=111">the original</a> on December 6, 2010.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Poet%27s+Corner+%E2%80%93+Fieralingue&rft_id=http%3A%2F%2Fwww.fieralingue.it%2Fmodules.php%3Fname%3DContent%26pa%3Dlist_pages_categories%26cid%3D111&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Travesty-111"><a href="#cite_ref-Travesty_111-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKennerO'Rourke1984" class="citation journal cs1">Kenner, Hugh; <a href="/wiki/Joseph_O%27Rourke_(professor)" title="Joseph O'Rourke (professor)">O'Rourke, Joseph</a> (November 1984). "A Travesty Generator for Micros". BYTE. 9 (12): 129–131, 449–469.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BYTE&rft.atitle=A+Travesty+Generator+for+Micros&rft.volume=9&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E129-%3C%2Fspan%3E131%2C+%3Cspan+class%3D%22nowrap%22%3E449-%3C%2Fspan%3E469&rft.date=1984-11&rft.aulast=Kenner&rft.aufirst=Hugh&rft.au=O%27Rourke%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> <li id="cite_note-Hartman-112"><a href="#cite_ref-Hartman_112-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartman1996" class="citation book cs1">Hartman, Charles (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/virtualmuseexper00hart">Virtual Muse: Experiments in Computer Poetry</a>. Hanover, NH: Wesleyan University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8195-2239-9" title="Special:BookSources/978-0-8195-2239-9"><bdi>978-0-8195-2239-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Virtual+Muse%3A+Experiments+in+Computer+Poetry&rft.place=Hanover%2C+NH&rft.pub=Wesleyan+University+Press&rft.date=1996&rft.isbn=978-0-8195-2239-9&rft.aulast=Hartman&rft.aufirst=Charles&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fvirtualmuseexper00hart&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=56" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>A. A. Markov (1906) "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga". Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2-ya seriya, tom 15, pp. 135–156.</li> <li>A. A. Markov (1971). "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wiley and Sons.</li> <li>Classical Text in Translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarkov2006" class="citation journal cs1">Markov, A. A. (2006). "An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains". Science in Context. 19 (4). Translated by Link, David: 591–600. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs0269889706001074">10.1017/s0269889706001074</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+in+Context&rft.atitle=An+Example+of+Statistical+Investigation+of+the+Text+Eugene+Onegin+Concerning+the+Connection+of+Samples+in+Chains&rft.volume=19&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E591-%3C%2Fspan%3E600&rft.date=2006&rft_id=info%3Adoi%2F10.1017%2Fs0269889706001074&rft.aulast=Markov&rft.aufirst=A.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"></li> <li>Leo Breiman (1992) [1968] Probability. Original edition published by Addison-Wesley; reprinted by <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-296-3" title="Special:BookSources/0-89871-296-3">0-89871-296-3</a>. (See Chapter 7)</li> <li><a href="/wiki/J._L._Doob" class="mw-redirect" title="J. L. Doob">J. L. Doob</a> (1953) Stochastic Processes. New York: John Wiley and Sons <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-52369-0" title="Special:BookSources/0-471-52369-0">0-471-52369-0</a>.</li> <li>S. P. Meyn and R. L. Tweedie (1993) Markov Chains and Stochastic Stability. London: Springer-Verlag <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-19832-6" title="Special:BookSources/0-387-19832-6">0-387-19832-6</a>. online: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100619010320/https://netfiles.uiuc.edu/meyn/www/spm_files/book.html">MCSS</a> . Second edition to appear, Cambridge University Press, 2009.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDynkin1965" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Eugene_Borisovich_Dynkin" class="mw-redirect" title="Eugene Borisovich Dynkin">Dynkin, Eugene Borisovich</a> (1965). <a rel="nofollow" class="external text" href="https://archive.org/details/markovprocesses0001dynk">Markov Processes</a>. Grundlehren der mathematischen Wissenschaften. Vol. I (121). Translated by Fabius, Jaap; Greenberg, Vida Lazarus; Maitra, Ashok Prasad; <a href="/wiki/Giandomenico_Majone" title="Giandomenico Majone">Majone, Giandomenico</a>. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-00031-1">10.1007/978-3-662-00031-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-00033-5" title="Special:BookSources/978-3-662-00033-5"><bdi>978-3-662-00033-5</bdi></a>. Title-No. 5104.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Markov+Processes&rft.place=Berlin&rft.series=Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer-Verlag&rft.date=1965&rft_id=info%3Adoi%2F10.1007%2F978-3-662-00031-1&rft.isbn=978-3-662-00033-5&rft.aulast=Dynkin&rft.aufirst=Eugene+Borisovich&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmarkovprocesses0001dynk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988">; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDynkin1965" class="citation book cs1 cs1-prop-long-vol"><a rel="nofollow" class="external text" href="https://archive.org/details/markovprocesses0002dynk">Markov Processes</a>. Grundlehren der mathematischen Wissenschaften. Vol. II (122). 1965. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-25360-1">10.1007/978-3-662-25360-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-23320-7" title="Special:BookSources/978-3-662-23320-7"><bdi>978-3-662-23320-7</bdi></a>. Title-No. 5105.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Markov+Processes&rft.series=Grundlehren+der+mathematischen+Wissenschaften&rft.date=1965&rft_id=info%3Adoi%2F10.1007%2F978-3-662-25360-1&rft.isbn=978-3-662-23320-7&rft.aulast=Dynkin&rft.aufirst=Eugene+Borisovich&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmarkovprocesses0002dynk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> (NB. This was originally published in Russian as Марковские процессы (Markovskiye protsessy) by <a href="/wiki/Fizmatgiz" class="mw-redirect" title="Fizmatgiz">Fizmatgiz</a> in 1963 and translated to English with the assistance of the author.)</li> <li>S. P. Meyn. Control Techniques for Complex Networks. Cambridge University Press, 2007. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88441-9" title="Special:BookSources/978-0-521-88441-9">978-0-521-88441-9</a>. Appendix contains abridged Meyn & Tweedie. online: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100619011046/https://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.html">CTCN</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBooth1967" class="citation book cs1">Booth, Taylor L. (1967). Sequential Machines and Automata Theory (1st ed.). New York, NY: John Wiley and Sons, Inc. Library of Congress Card Catalog Number 67-25924.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sequential+Machines+and+Automata+Theory&rft.place=New+York%2C+NY&rft.edition=1st&rft.pub=John+Wiley+and+Sons%2C+Inc.&rft.date=1967&rft.aulast=Booth&rft.aufirst=Taylor+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> ] Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes pp. 449ff. Discusses Z-transforms, D transforms in their context.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKemenyHazleton_MirkilJ._Laurie_SnellGerald_L._Thompson1959" class="citation book cs1">Kemeny, John G.; Hazleton Mirkil; J. Laurie Snell; Gerald L. Thompson (1959). <a rel="nofollow" class="external text" href="https://archive.org/details/finitemathematic0000keme_h5g0">Finite Mathematical Structures</a> (1st ed.). Englewood Cliffs, NJ: Prentice-Hall, Inc. Library of Congress Card Catalog Number 59-12841.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Mathematical+Structures&rft.place=Englewood+Cliffs%2C+NJ&rft.edition=1st&rft.pub=Prentice-Hall%2C+Inc.&rft.date=1959&rft.aulast=Kemeny&rft.aufirst=John+G.&rft.au=Hazleton+Mirkil&rft.au=J.+Laurie+Snell&rft.au=Gerald+L.+Thompson&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffinitemathematic0000keme_h5g0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"> Classical text. cf Chapter 6 Finite Markov Chains pp. 384ff.</li> <li><a href="/wiki/John_G._Kemeny" title="John G. Kemeny">John G. Kemeny</a> & <a href="/wiki/J._Laurie_Snell" title="J. Laurie Snell">J. Laurie Snell</a> (1960) Finite Markov Chains, D. van Nostrand Company <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-442-04328-7" title="Special:BookSources/0-442-04328-7">0-442-04328-7</a></li> <li>E. Nummelin. "General irreducible Markov chains and non-negative operators". Cambridge University Press, 1984, 2004. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-60494-X" title="Special:BookSources/0-521-60494-X">0-521-60494-X</a></li> <li>Seneta, E. Non-negative matrices and Markov chains. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-29765-1" title="Special:BookSources/978-0-387-29765-1">978-0-387-29765-1</a></li> <li><a href="/wiki/Kishor_S._Trivedi" title="Kishor S. Trivedi">Kishor S. Trivedi</a>, Probability and Statistics with Reliability, Queueing, and Computer Science Applications, John Wiley & Sons, Inc. New York, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-33341-7" title="Special:BookSources/0-471-33341-7">0-471-33341-7</a>.</li> <li>K. S. Trivedi and R.A.Sahner, SHARPE at the age of twenty-two, vol. 36, no. 4, pp. 52–57, ACM SIGMETRICS Performance Evaluation Review, 2009.</li> <li>R. A. Sahner, K. S. Trivedi and A. Puliafito, Performance and reliability analysis of computer systems: an example-based approach using the SHARPE software package, Kluwer Academic Publishers, 1996. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-9650-2" title="Special:BookSources/0-7923-9650-2">0-7923-9650-2</a>.</li> <li>G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi, Queueing Networks and Markov Chains, John Wiley, 2nd edition, 2006. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-9650-5" title="Special:BookSources/978-0-7923-9650-5">978-0-7923-9650-5</a>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Markov_chain&action=edit&section=57" title="Edit section: External links">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Markov_chain">"Markov chain"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Markov+chain&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMarkov_chain&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMarkov+chain" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf">Markov Chains chapter in American Mathematical Society's introductory probability book</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080522131917/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf">Archived</a> 2008-05-22 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=o-jdJxXL_W4">Introduction to Markov Chains</a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a></li> <li><a rel="nofollow" class="external text" href="http://setosa.io/blog/2014/07/26/markov-chains/index.html">A visual explanation of Markov Chains</a></li> <li><a rel="nofollow" class="external text" href="http://www.alpha60.de/research/markov/DavidLink_AnExampleOfStatistical_MarkovTrans_2007.pdf">Original paper by A.A Markov (1913): An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains (translated from Russian)</a></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Stochastic_processes496" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Stochastic_processes" title="Template:Stochastic processes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Stochastic_processes" title="Template talk:Stochastic processes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Stochastic_processes" title="Special:EditPage/Template:Stochastic processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a class="mw-selflink selflink">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a href="/wiki/G-network" title="G-network">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Control theory</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li> <li><a href="/wiki/Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li> <li><a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li> <li><a href="/wiki/Category:Stochastic_processes" title="Category:Stochastic processes">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q176645#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1109" style="padding:3px"><table class="nowraplinks hlist 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