CINXE.COM

<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="8juCM1l4z3JOmhA6ri8d6svGVsCSkvQIgyp8oNudj1inHvuqZevhnW7XDMT6lLyKrxAvmiBo901qefhkpVJpCQ" /> <meta name="citation_title" content="q-Wiener and related processes. A continuous time generalization of Bryc processes" /> <meta name="citation_publication_date" content="2005/07/14" /> <meta name="citation_journal_title" content="arXiv (Cornell University)" /> <meta name="citation_author" content="Paweł J Szabłowski" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes" /> <meta name="twitter:title" content="q-Wiener and related processes. A continuous time generalization of Bryc processes" /> <meta name="twitter:description" content="We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in" /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes" /> <meta property="og:title" content="q-Wiener and related processes. A continuous time generalization of Bryc processes" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in" /> <meta property="article:author" content="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski" /> <meta name="description" content="We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in" /> <title>(PDF) q-Wiener and related processes. A continuous time generalization of Bryc processes</title> <link rel="canonical" href="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '0b8ad487192af8d1cd4b80bd34002cf444c419e0'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1741350073000); window.Aedu.timeDifference = new Date().getTime() - 1741350073000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","author":[{"@context":"https://schema.org","@type":"Person","name":"Paweł J Szabłowski","url":"https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski","sameAs":[]}],"contributor":[],"dateCreated":"2024-05-30","dateModified":"2024-11-23","datePublished":"2005-07-14","headline":"q-Wiener and related processes. A continuous time generalization of Bryc processes","image":"https://attachments.academia-assets.com/115484111/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematics","Generalization","Gaussian distribution","Boolean Satisfiability","Markov Process","Correlation function","Discrete time"],"publication":"arXiv (Cornell University)","publisher":{"@context":"https://schema.org","@type":"Organization","name":"Cornell University"},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"racjonalista"}],"thumbnailUrl":"https://attachments.academia-assets.com/115484111/thumbnails/1.jpg","url":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 32px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 30px;--type-sans-serif-lg-line-height: 36px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 30px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 24px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 24px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 18px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 32px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 20px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 32px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 40px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 24px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 26px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 48px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 52px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 48px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 58px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 80px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 64px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}@media(min-width: 568px)and (max-width: 1279px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 42px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 32px;--type-sans-serif-lg-line-height: 36px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 34px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 28px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 25px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 20px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 30px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 24px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 40px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 48px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 28px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 32px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 58px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 68px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 74px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 82px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 104px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 80px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}@media(min-width: 1280px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 42px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 32px;--type-sans-serif-lg-line-height: 38px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 34px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 28px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 25px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 20px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 30px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 24px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 40px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 48px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 28px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 32px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 58px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 68px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 74px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 82px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 152px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 80px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}</style><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/single_work_page/loswp-369df59c8124f51e480079869c157eba45d12fcd3f65e2af6d7f44b33daec50d.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/body-170d1319f0e354621e81ca17054bb147da2856ec0702fe440a99af314a6338c5.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/button-8c9ae4b5c8a2531640c354d92a1f3579c8ff103277ef74913e34c8a76d4e6c00.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/text_button-d1941ab08e91e29ee143084c4749da4aaffa350a2ac6eec2306b1d7a352d911a.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/heading-95367dc03b794f6737f30123738a886cf53b7a65cdef98a922a98591d60063e3.css" /><link crossorigin="" href="https://fonts.gstatic.com/" rel="preconnect" /><link href="https://fonts.googleapis.com/css2?family=DM+Sans:ital,opsz,wght@0,9..40,100..1000;1,9..40,100..1000&family=Gupter:wght@400;500;700&family=IBM+Plex+Mono:wght@300;400&family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20,400,0,0&display=swap" rel="stylesheet" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/common-57f9da13cef3fd4e2a8b655342c6488eded3e557e823fe67571f2ac77acd7b6f.css" /> </head> <body> <div id='react-modal'></div> <div class="js-upgrade-ie-banner" style="display: none; text-align: center; padding: 8px 0; background-color: #ebe480;"><p style="color: #000; font-size: 12px; margin: 0 0 4px;">Academia.edu no longer supports Internet Explorer.</p><p style="color: #000; font-size: 12px; margin: 0;">To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to <a href="https://www.academia.edu/upgrade-browser">upgrade your browser</a>.</p></div><script>// Show this banner for all versions of IE if (!!window.MSInputMethodContext || /(MSIE)/.test(navigator.userAgent)) { document.querySelector('.js-upgrade-ie-banner').style.display = 'block'; }</script> <div class="bootstrap login"><div class="modal fade login-modal" id="login-modal"><div class="login-modal-dialog modal-dialog"><div class="modal-content"><div class="modal-header"><button class="close close" data-dismiss="modal" type="button"><span aria-hidden="true">×</span><span class="sr-only">Close</span></button><h4 class="modal-title text-center"><strong>Log In</strong></h4></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><button class="btn btn-fb btn-lg btn-block btn-v-center-content" id="login-facebook-oauth-button"><svg style="float: left; width: 19px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="facebook-square" class="svg-inline--fa fa-facebook-square fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M400 32H48A48 48 0 0 0 0 80v352a48 48 0 0 0 48 48h137.25V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14 0 55.52 4.84 55.52 4.84v61h-31.27c-30.81 0-40.42 19.12-40.42 38.73V256h68.78l-11 71.69h-57.78V480H400a48 48 0 0 0 48-48V80a48 48 0 0 0-48-48z"></path></svg><small><strong>Log in</strong> with <strong>Facebook</strong></small></button><br /><button class="btn btn-google btn-lg btn-block btn-v-center-content" id="login-google-oauth-button"><svg style="float: left; width: 22px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="google-plus" class="svg-inline--fa fa-google-plus fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256,8C119.1,8,8,119.1,8,256S119.1,504,256,504,504,392.9,504,256,392.9,8,256,8ZM185.3,380a124,124,0,0,1,0-248c31.3,0,60.1,11,83,32.3l-33.6,32.6c-13.2-12.9-31.3-19.1-49.4-19.1-42.9,0-77.2,35.5-77.2,78.1S142.3,334,185.3,334c32.6,0,64.9-19.1,70.1-53.3H185.3V238.1H302.2a109.2,109.2,0,0,1,1.9,20.7c0,70.8-47.5,121.2-118.8,121.2ZM415.5,273.8v35.5H380V273.8H344.5V238.3H380V202.8h35.5v35.5h35.2v35.5Z"></path></svg><small><strong>Log in</strong> with <strong>Google</strong></small></button><br /><style type="text/css">.sign-in-with-apple-button { width: 100%; height: 52px; border-radius: 3px; border: 1px solid black; cursor: pointer; } .sign-in-with-apple-button > div { margin: 0 auto; / This centers the Apple-rendered button horizontally }</style><script src="https://appleid.cdn-apple.com/appleauth/static/jsapi/appleid/1/en_US/appleid.auth.js" type="text/javascript"></script><div class="sign-in-with-apple-button" data-border="false" data-color="white" id="appleid-signin"><span   ="Sign Up with Apple" class="u-fs11"></span></div><script>AppleID.auth.init({ clientId: 'edu.academia.applesignon', scope: 'name email', redirectURI: 'https://www.academia.edu/sessions', state: "0e67c631fb61ef801dee3dd409b79c4b4d8830817bba557eb3a7688ea4583d70", });</script><script>// Hacky way of checking if on fast loswp if (window.loswp == null) { (function() { const Google = window?.Aedu?.Auth?.OauthButton?.Login?.Google; const Facebook = window?.Aedu?.Auth?.OauthButton?.Login?.Facebook; if (Google) { new Google({ el: '#login-google-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } if (Facebook) { new Facebook({ el: '#login-facebook-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } })(); }</script></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><div class="hr-heading login-hr-heading"><span class="hr-heading-text">or</span></div></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><form class="js-login-form" action="https://www.academia.edu/sessions" accept-charset="UTF-8" method="post"><input type="hidden" name="authenticity_token" value="ZRqVWLxiPh2x_5fZxSRT5QgwLGlvWhHvvutRlI-N0TwwP-zBgPEQ8pGyiyeRn_KFbOZVM92gEqpXuNVQ8UI3bQ" autocomplete="off" /><div class="form-group"><label class="control-label" for="login-modal-email-input" style="font-size: 14px;">Email</label><input class="form-control" id="login-modal-email-input" name="login" type="email" /></div><div class="form-group"><label class="control-label" for="login-modal-password-input" style="font-size: 14px;">Password</label><input class="form-control" id="login-modal-password-input" name="password" type="password" /></div><input type="hidden" name="post_login_redirect_url" id="post_login_redirect_url" value="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes" autocomplete="off" /><div class="checkbox"><label><input type="checkbox" name="remember_me" id="remember_me" value="1" checked="checked" /><small style="font-size: 12px; margin-top: 2px; display: inline-block;">Remember me on this computer</small></label></div><br><input type="submit" name="commit" value="Log In" class="btn btn-primary btn-block btn-lg js-login-submit" data-disable-with="Log In" /></br></form><script>typeof window?.Aedu?.recaptchaManagedForm === 'function' && window.Aedu.recaptchaManagedForm( document.querySelector('.js-login-form'), document.querySelector('.js-login-submit') );</script><small style="font-size: 12px;"><br />or <a data-target="#login-modal-reset-password-container" data-toggle="collapse" href="javascript:void(0)">reset password</a></small><div class="collapse" id="login-modal-reset-password-container"><br /><div class="well margin-0x"><form class="js-password-reset-form" action="https://www.academia.edu/reset_password" accept-charset="UTF-8" method="post"><input type="hidden" name="authenticity_token" value="SuPLMuds99RUfo0Sca66xGcfbwrxI6ktLqiKOC8YaqYfxrKr2__ZO3QzkewlFRukA8kWUEPZqmjH-w78UdeM9w" autocomplete="off" /><p>Enter the email address you signed up with and we'll email you a reset link.</p><div class="form-group"><input class="form-control" name="email" type="email" /></div><input class="btn btn-primary btn-block g-recaptcha js-password-reset-submit" data-sitekey="6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj" type="submit" value="Email me a link" /></form></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/collapse-45805421cf446ca5adf7aaa1935b08a3a8d1d9a6cc5d91a62a2a3a00b20b3e6a.js"], function() { // from javascript_helper.rb $("#login-modal-reset-password-container").on("shown.bs.collapse", function() { $(this).find("input[type=email]").focus(); }); }); </script> </div></div></div><div class="modal-footer"><div class="text-center"><small style="font-size: 12px;">Need an account? <a rel="nofollow" href="https://www.academia.edu/signup">Click here to sign up</a></small></div></div></div></div></div></div><script>// If we are on subdomain or non-bootstrapped page, redirect to login page instead of showing modal (function(){ if (typeof $ === 'undefined') return; var host = window.location.hostname; if ((host === $domain || host === "www."+$domain) && (typeof $().modal === 'function')) { $("#nav_log_in").click(function(e) { // Don't follow the link and open the modal e.preventDefault(); $("#login-modal").on('shown.bs.modal', function() { $(this).find("#login-modal-email-input").focus() }).modal('show'); }); } })()</script> <div id="fb-root"></div><script>window.fbAsyncInit = function() { FB.init({ appId: "2369844204", version: "v8.0", status: true, cookie: true, xfbml: true }); // Additional initialization code. if (window.InitFacebook) { // facebook.ts already loaded, set it up. window.InitFacebook(); } else { // Set a flag for facebook.ts to find when it loads. window.academiaAuthReadyFacebook = true; } };</script> <div id="google-root"></div><script>window.loadGoogle = function() { if (window.InitGoogle) { // google.ts already loaded, set it up. window.InitGoogle("331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"); } else { // Set a flag for google.ts to use when it loads. window.GoogleClientID = "331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"; } };</script> <div class="header--container" id="main-header-container"><div class="header--inner-container header--inner-container-ds2"><div class="header-ds2--left-wrapper"><div class="header-ds2--left-wrapper-inner"><a data-main-header-link-target="logo_home" href="https://www.academia.edu/"><img class="hide-on-desktop-redesign" style="height: 24px; width: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015-A.svg" width="24" height="24" /><img width="145.2" height="18" class="hide-on-mobile-redesign" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015.svg" /></a><div class="header--search-container header--search-container-ds2"><form class="js-SiteSearch-form select2-no-default-pills" action="https://www.academia.edu/search" accept-charset="UTF-8" method="get"><svg style="width: 14px; height: 14px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="search" class="header--search-icon svg-inline--fa fa-search fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M505 442.7L405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9 0 208 0S0 93.1 0 208s93.1 208 208 208c48.3 0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9 0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7 0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7 0 128 57.2 128 128 0 70.7-57.2 128-128 128z"></path></svg><input class="header--search-input header--search-input-ds2 js-SiteSearch-form-input" data-main-header-click-target="search_input" name="q" placeholder="Search" type="text" /></form></div></div></div><nav class="header--nav-buttons header--nav-buttons-ds2 js-main-nav"><button class="ds2-5-button ds2-5-button--secondary js-header-login-url header-button-ds2 header-login-ds2 hide-on-mobile-redesign react-login-modal-opener" data-signup-modal="{"location":"login-button--header"}" rel="nofollow">Log In</button><button class="ds2-5-button ds2-5-button--secondary header-button-ds2 hide-on-mobile-redesign react-login-modal-opener" data-signup-modal="{"location":"signup-button--header"}" rel="nofollow">Sign Up</button><button class="header--hamburger-button header--hamburger-button-ds2 hide-on-desktop-redesign js-header-hamburger-button"><div class="icon-bar"></div><div class="icon-bar" style="margin-top: 4px;"></div><div class="icon-bar" style="margin-top: 4px;"></div></button></nav></div><ul class="header--dropdown-container js-header-dropdown"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/login" rel="nofollow">Log In</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/signup" rel="nofollow">Sign Up</a></li><li class="header--dropdown-row js-header-dropdown-expand-button"><button class="header--dropdown-button">more<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-down" class="header--dropdown-button-icon svg-inline--fa fa-caret-down fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M31.3 192h257.3c17.8 0 26.7 21.5 14.1 34.1L174.1 354.8c-7.8 7.8-20.5 7.8-28.3 0L17.2 226.1C4.6 213.5 13.5 192 31.3 192z"></path></svg></button></li><li><ul class="header--expanded-dropdown-container"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/about">About</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/press">Press</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/documents">Papers</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/terms">Terms</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/privacy">Privacy</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/copyright">Copyright</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/hiring"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="header--dropdown-row-icon svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg>We're Hiring!</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://support.academia.edu/hc/en-us"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="header--dropdown-row-icon svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg>Help Center</a></li><li class="header--dropdown-row js-header-dropdown-collapse-button"><button class="header--dropdown-button">less<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-up" class="header--dropdown-button-icon svg-inline--fa fa-caret-up fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M288.662 352H31.338c-17.818 0-26.741-21.543-14.142-34.142l128.662-128.662c7.81-7.81 20.474-7.81 28.284 0l128.662 128.662c12.6 12.599 3.676 34.142-14.142 34.142z"></path></svg></button></li></ul></li></ul></div> <script src="//a.academia-assets.com/assets/webpack_bundles/fast_loswp-bundle-08cf32c759029a9081b84345798e35bf76494c74d8d264829dd9293771a1c785.js" defer="defer"></script><script>window.loswp = {}; window.loswp.author = 55567278; window.loswp.bulkDownloadFilterCounts = {}; window.loswp.hasDownloadableAttachment = true; window.loswp.hasViewableAttachments = true; // TODO: just use routes for this window.loswp.loginUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F120282236%2Fq_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes%3Fauto%3Ddownload"; window.loswp.translateUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F120282236%2Fq_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes%3Fshow_translation%3Dtrue"; window.loswp.previewableAttachments = [{"id":115484111,"identifier":"Attachment_115484111","shouldShowBulkDownload":false}]; window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":120282236,"created_at":"2024-05-30T11:35:36.268-07:00","from_world_paper_id":255360739,"updated_at":"2024-11-23T05:54:19.828-08:00","_data":{"publisher":"Cornell University","grobid_abstract":"We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| \u003c 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| \u003c 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.","publication_date":"2005,7,14","publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":"115484111"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"q-Wiener and related processes. A continuous time generalization of Bryc processes","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true,"seo_quality":null}}["work"]; window.loswp.workCoauthors = [55567278]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":115484111,"attachmentType":"pdf"}"><img alt="First page of “q-Wiener and related processes. A continuous time generalization of Bryc processes”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/115484111/mini_magick20240802-1-yaannt.png?1722574687" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">q-Wiener and related processes. A continuous time generalization of Bryc processes</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="55567278" href="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski"><img alt="Profile image of Paweł J Szabłowski" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Paweł J Szabłowski</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2005, arXiv (Cornell University)</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">25 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 120282236; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=120282236"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X = (Xt) t≥0 called q−Wiener) depends on one parameter q ∈ (−1, 1] and of the second one (say Y = (Yt) t∈R called (α, q)− Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for |q| < 1 and ∀n ≥ 1 : t n/2 Hn Xt/ √ t|q , where (Hn) n≥0 are the so called q−Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein-Uhlenbeck process. For q = 1 it is a classical OU process. For |q| < 1 it is also stationary with correlation function equal to exp(−α|t − s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.</p><div class="ds-work-card--button-container"><div class="primary-buttons cite-share-variant"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":115484111,"attachmentType":"pdf","workUrl":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span>See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":115484111,"attachmentType":"pdf","workUrl":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button><div id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes","location":"/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes","scheme":"https","host":"www.academia.edu","port":null,"pathname":"/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="CitationButton" data-props="{"entity_id":120282236,"citations":[{"name":"MLA","citation":"Szabłowski, Paweł J. “q-Wiener and Related Processes. A Continuous Time Generalization of Bryc Processes.” ArXiv (Cornell University), Cornell University, 2005."},{"name":"APA","citation":"Szabłowski, P. J. (2005). q-Wiener and related processes. A continuous time generalization of Bryc processes. ArXiv (Cornell University)."},{"name":"Chicago","citation":"Szabłowski, Paweł J. “q-Wiener and Related Processes. A Continuous Time Generalization of Bryc Processes.” ArXiv (Cornell University), 2005."},{"name":"Vancouver","citation":"Szabłowski PJ. q-Wiener and related processes. A continuous time generalization of Bryc processes. arXiv (Cornell University). 2005; "},{"name":"Harvard","citation":"Szabłowski, P. J. (2005) “q-Wiener and related processes. A continuous time generalization of Bryc processes,” arXiv (Cornell University). Cornell University."}],"source":"swp_citation_modal","buttonClassName":"ds-work-card--citation-button"}" data-trace="false" data-dom-id="CitationButton-react-component-00621378-015c-4d03-acc5-d6b924fb8643"></div> <div id="CitationButton-react-component-00621378-015c-4d03-acc5-d6b924fb8643"></div> </div><div class="share-and-save-buttons"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"bookmark-button--work-card","attachmentId":115484111,"attachmentType":"pdf","workUrl":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes"}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">bookmark</span><span class="ds2-5-text-link__content">Save to Library</span></button><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"share-button--work-card","attachmentId":115484111,"attachmentType":"pdf","workUrl":"https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes"}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">share</span><span class="ds2-5-text-link__content">Share</span></button></div></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="120282236" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. If the signup banner trigger (a 242px-high* invisible div underneath the 'See Full PDF' / 'Download PDF' buttons) // is already fully scrolled above the viewport, show the banner by default // 2. If the signup banner trigger is fully visible, show the banner // 3. If the signup banner trigger has even a few pixels scrolled below the viewport, hide the banner // // * 242px is the empirically determined height of the signup banner. It's better to be a bit taller than // necessary than too short, so it's fine that the mobile (small breakpoint) banner is shorter. // First check session storage for the signup banner's visibility state const signupBannerHidden = sessionStorage.getItem('ds-signup-banner-hidden'); if (signupBannerHidden === 'true') { return; } const signupBanner = document.querySelector('.ds-signup-banner'); const signupBannerTrigger = document.querySelector('.ds-signup-banner-trigger'); if (!signupBannerTrigger) { window.Sentry.captureMessage("Signup banner trigger not found"); return; } let footerShown = false; window.addEventListener('load', () => { const rect = signupBannerTrigger.getBoundingClientRect(); // If page loaded up already scrolled below the trigger (via scroll restoration), show the banner by default if (rect.bottom < 0) { footerShown = true; signupBanner.classList.add('ds-signup-banner-visible'); } }); // Wait for trigger to fully enter viewport before showing banner (ensures PDF CTAs are never covered by banner) const observer = new IntersectionObserver((entries) => { entries.forEach(entry => { if (entry.isIntersecting && !footerShown) { footerShown = true; signupBanner.classList.add('ds-signup-banner-visible'); } else if (!entry.isIntersecting && footerShown) { if (signupBannerTrigger.getBoundingClientRect().bottom > 0) { footerShown = false; signupBanner.classList.remove('ds-signup-banner-visible'); } } }); }); observer.observe(signupBannerTrigger); // Set up signup banner close button event handler: const signupBannerCloseButton = document.querySelector('#ds-signup-banner-close-button'); signupBannerCloseButton.addEventListener('click', () => { signupBanner.classList.remove('ds-signup-banner-visible'); observer.unobserve(signupBannerTrigger); // Store the signup banner's visibility state in session storage sessionStorage.setItem('ds-signup-banner-hidden', 'true'); }); })();</script></div></div></div><style>@media screen and (min-width: 568px) and (max-width: 636px) { .ds-work-card--button-container .primary-buttons .ds2-5-button, .ds-work-card--button-container .primary-buttons .ds-work-card--citation-button { font-size: var(--font-size-small); line-height: var(--line-height-small); padding: 0 var(--buttons-small-buttons-l-r-padding, 12px); height: var(--buttons-small-buttons-height, 44px); gap: var(--buttons-small-buttons-gap, 8px); border-radius: var(--buttons-small-buttons-corner-radius, 8px); }}</style><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="115484111" data-landing_url="https://www.academia.edu/120282236/q_Wiener_and_related_processes_A_continuous_time_generalization_of_Bryc_processes" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="70984193" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/70984193/q_WIENER_AND_%CE%B1_q_ORNSTEIN_UHLENBECK_PROCESSES_A_GENERALIZATION_OF_KNOWN_PROCESSES">q−WIENER AND (α, q)−ORNSTEIN-UHLENBECK PROCESSES. A GENERALIZATION OF KNOWN PROCESSES</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="55567278" href="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski">Paweł J Szabłowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2012</p><p class="ds-related-work--abstract ds2-5-body-sm">We collect and prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein–Uhlenbeck (OU) processes. Although processes considered in this paper were defined either in a noncommutative probability context or through quadratic harnesses we define them once more as a “continuous time” generalization of a simple, symmetric, discrete-time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite-dimensional distributions of the first one (say X =(Xt)t0 called q-Wiener) depend on one parameter q ∈ (−1, 1], and those of the second one (say Y =(Yt)t∈R called (α, q)-Ornstein–Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles the Wiener process in the sense that for q = 1 it is a Wiener process but also that for |q| < 1 and ∀n 1: tn/2Hn(Xt/√t | q), where (Hn)n0 are the so-called q-Hermite polynomials, are martingales. However, it neither has independent increments not allows continuous sample path modification. The second one resembles the OU process. For q = 1 it is a classical OU process. For |q| < 1 it is also stationary with a correlation function equal to exp(−α|t−s|) and has many properties resembling those of its classical version. We think that these processes are fascinating objects to study posing many interesting, open questions.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"q−WIENER AND (α, q)−ORNSTEIN-UHLENBECK PROCESSES. A GENERALIZATION OF KNOWN PROCESSES","attachmentId":80551726,"attachmentType":"pdf","work_url":"https://www.academia.edu/70984193/q_WIENER_AND_%CE%B1_q_ORNSTEIN_UHLENBECK_PROCESSES_A_GENERALIZATION_OF_KNOWN_PROCESSES","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/70984193/q_WIENER_AND_%CE%B1_q_ORNSTEIN_UHLENBECK_PROCESSES_A_GENERALIZATION_OF_KNOWN_PROCESSES"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="29407118" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/29407118/_q_Wiener_and_alpha_q_Ornstein_Uhlenbeck_Processes_A_Generalization_of_Known_Processes">$q$-Wiener and $(\alpha,q)$-Ornstein--Uhlenbeck Processes. A Generalization of Known Processes</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="55567278" href="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski">Paweł J Szabłowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Theory of Probability & Its Applications, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">We collect and prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein-Uhlenbeck (OU) processes. Although processes considered in this paper were defined either in a noncommutative probability context or through quadratic harnesses we define them once more as a "continuous time" generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X =(Xt) t 0 called q-Wiener) depend on one parameter q ∈ (−1, 1], and those of the second one (say Y =(Yt) t∈R called (α, q)-Ornstein-Uhlenbeck) on two parameters (α, q) ∈ (0, ∞)×(−1, 1]. The first one resembles the Wiener process in the sense that for q = 1 it is a Wiener process but also that for |q| < 1 and ∀n 1: t n/2 Hn(Xt/ √ t | q), where (Hn) n 0 are the so-called q-Hermite polynomials, are martingales. However, it neither has independent increments not allows continuous sample path modification. The second one resembles the OU process. For q = 1 it is a classical OU process. For |q| < 1 it is also stationary with correlation function equal to exp(−α|t−s|) and has many properties resembling those of its classical version. We think that these processes are fascinating objects to study posing many interesting, open questions. 635 several (exactly five) conditions on the covariance function and on the first and second conditional moments. Those are the so-called quadratic harnesses characterized by five parameters. Under resulting assumptions they proved that these processes are Markov and also stated several properties of the families of polynomials that orthogonalize the transitional and the one-dimensional probabilities. They gave also several examples illustrating the developed theory. One of the processes considered by them is the so-called q-Brownian process. Four of five possible parameters are equal to zero and the fifth one can be identified with the parameter q considered in this paper. As far as the one-dimensional probabilities and the transitional probabilities are concerned, this process is identical to the q-Wiener process introduced and analyzed in this paper. Bryc and coworkers did not, however, work on the properties of the q-Brownian process. It appeared as a by-product of their interest in quadratic harnesses. Bryc, Matysiak, and Weso lowski were mostly interested in the general problem of existence of quadratic harnesses. That is why the (α, q)-OU process has not appeared in their work.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"$q$-Wiener and $(\\alpha,q)$-Ornstein--Uhlenbeck Processes. A Generalization of Known Processes","attachmentId":49848240,"attachmentType":"pdf","work_url":"https://www.academia.edu/29407118/_q_Wiener_and_alpha_q_Ornstein_Uhlenbeck_Processes_A_Generalization_of_Known_Processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/29407118/_q_Wiener_and_alpha_q_Ornstein_Uhlenbeck_Processes_A_Generalization_of_Known_Processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="77936616" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/77936616/q_Wiener_%CE%B1_q_Ornstein_Uhlenbeck_processes_A_generalization_of_known_processes">q-Wiener (α,q)- Ornstein-Uhlenbeck processes. A generalization of known processes</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="55567278" href="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski">Paweł J Szabłowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2010</p><p class="ds-related-work--abstract ds2-5-body-sm">We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say &#39;continuous time &#39; generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X=(X_t)_t≥0 called q-Wiener) depends on one parameter q∈(-1,1] and of the second one (say Y=(Y_t)_t called (α,q)- Ornstein--Uhlenbeck) on two parameters (α,q)∈(0,∞)×(-1,1]. The first one resembles Wiener process in the sense that for q=1 it is Wiener process but also that for |q|&lt;1 and ≥1: t^n/2H_n(X_t/|q), where (H_n)_n≥0 are the so called q-Hermite polynomials, are martingales. It does not have however neith...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"q-Wiener (α,q)- Ornstein-Uhlenbeck processes. A generalization of known processes","attachmentId":85157565,"attachmentType":"pdf","work_url":"https://www.academia.edu/77936616/q_Wiener_%CE%B1_q_Ornstein_Uhlenbeck_processes_A_generalization_of_known_processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/77936616/q_Wiener_%CE%B1_q_Ornstein_Uhlenbeck_processes_A_generalization_of_known_processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="65075671" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/65075671/q_Wiener_and_related_processes_A_Bryc_processes_continuous_time_generalization">q-Wiener and related processes. A Bryc processes continuous time generalization</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="55567278" href="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski">Paweł J Szabłowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2005</p><p class="ds-related-work--abstract ds2-5-body-sm">We define two Markov processes. The finite dimensional distributions of the first one (say X = (Xt)t≥0) depend on one parameter q ∈ (−1, 1 &gt; and of the second one (say Y = (Yt)t∈R) on two parameters (q, α) ∈ (−1, 1 &gt; ×(0,∞). The first one resembles Wiener process in the sense that for q = 1 it is Wiener process but also that for q &lt; 1 and ∀n ≥ 1 tHn (</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"q-Wiener and related processes. A Bryc processes continuous time generalization","attachmentId":76820939,"attachmentType":"pdf","work_url":"https://www.academia.edu/65075671/q_Wiener_and_related_processes_A_Bryc_processes_continuous_time_generalization","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/65075671/q_Wiener_and_related_processes_A_Bryc_processes_continuous_time_generalization"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="23157577" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/23157577/CONSTRUCTION_OF_NEW_CONTINUOUS_TIME_STOCHASTIC_PROCESSES">CONSTRUCTION OF NEW CONTINUOUS TIME STOCHASTIC PROCESSES</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="41812569" href="https://oakton.academia.edu/JerzyFilus">Jerzy K Filus</a></div><p class="ds-related-work--abstract ds2-5-body-sm">The class of pseudoaffine (or more general 'triangular') R n  R n (n  2) transformations, used in our previous works in the process of constructing n-variate pdfs (such as, for example, pseudonormal), is extended, as n  , to infinite dimension spaces. This transition opens the way for construction of many new stochastic processes with discrete time. As a next step, a transition to continuous time processes enriches the underlying theory. The so obtained stochastic processes reveal interesting analytic properties. Some of the newly obtained processes are Markovian or k-th order Markovian. As a central result, the Wiener process is obtained along with its new generalization. A subclass, of the so obtained class of extended Wiener stochastic processes, is also shown to be alternatively described by a corresponding class of parabolic form partial differential equations with variable coefficients. That class of differential equations is an extension of the well known class of Einstein's parabolic differential equations related to the original Wiener process. There is a one to one correspondence between each particular extended Wiener process belonging to the defined processes subclass, and related extended Einstein's partial differential equation.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"CONSTRUCTION OF NEW CONTINUOUS TIME STOCHASTIC PROCESSES","attachmentId":43641870,"attachmentType":"pdf","work_url":"https://www.academia.edu/23157577/CONSTRUCTION_OF_NEW_CONTINUOUS_TIME_STOCHASTIC_PROCESSES","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/23157577/CONSTRUCTION_OF_NEW_CONTINUOUS_TIME_STOCHASTIC_PROCESSES"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="117032140" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/117032140/A_new_class_of_stochastic_processes_with_great_potential_for_interesting_applications">A new class of stochastic processes with great potential for interesting applications</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="150847270" href="https://ustmasuku.academia.edu/OctaveMoutsinga">Octave Moutsinga</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2022</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper contributes to the study of a new and remarkable family of stochastic processes that we will term class Σ r (H). This class is potentially interesting because it unifies the study of two known classes: the class (Σ) and the class M(H). In other words, we consider the stochastic processes X which decompose as X = m+v+A, where m is a local martingale, v and A are finite variation processes such that dA is carried by {t ≥ 0 : X t = 0} and the support of dv is H, the set of zeros of some continuous martingale D. First, we introduce a general framework. Thus, we provide some examples of elements of the new class and present some properties. Second, we provide a series of characterization results. Afterwards, we derive some representation results which permit to recover a process of the class Σ r (H) from its final value and of the honest times g = sup{t ≥ 0 : X t = 0} and γ = sup H. In final, we investigate an interesting application with processes presently studied. More precisely, we construct solutions for skew Brownian motion equations using stochastic processes of the class Σ r (H).</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A new class of stochastic processes with great potential for interesting applications","attachmentId":112999372,"attachmentType":"pdf","work_url":"https://www.academia.edu/117032140/A_new_class_of_stochastic_processes_with_great_potential_for_interesting_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/117032140/A_new_class_of_stochastic_processes_with_great_potential_for_interesting_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="51879155" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/51879155/Some_contributions_to_the_study_of_stochastic_processes_of_the_classes_and">Some contributions to the study of stochastic processes of the classes and</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51424466" href="https://ucam-ma.academia.edu/YOuknine">Youssef Ouknine</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Stochastics</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper consists of two independent parts. In the first one, we contribute to the study of the class (Σ). For instance, we provide a new way to characterize stochastic processes of this class. We also present some new properties and solve the Bachelier equation. In the second part, we study the class of stochastic processes Σ(H). This class was introduced in [6] where from tools of the theory of martingales with respect to a signed measure of [19], the authors provide a general framework and methods for dealing with processes of this class. In this work, after developing some new properties, we embed a non-atomic measure ν in X, a process of the class Σ(H). More precisely, we find a stopping time T < ∞ such that the law of X T is ν.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Some contributions to the study of stochastic processes of the classes and","attachmentId":69403514,"attachmentType":"pdf","work_url":"https://www.academia.edu/51879155/Some_contributions_to_the_study_of_stochastic_processes_of_the_classes_and","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/51879155/Some_contributions_to_the_study_of_stochastic_processes_of_the_classes_and"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="56288608" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/56288608/Stochastic_processes_with_applications">Stochastic processes with applications</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="8942455" href="https://independent.academia.edu/EdWaymire">Ed Waymire</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Computation and Simulation, 2013</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Stochastic processes with applications","attachmentId":71748125,"attachmentType":"pdf","work_url":"https://www.academia.edu/56288608/Stochastic_processes_with_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/56288608/Stochastic_processes_with_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="101323264" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/101323264/Processes_">Processes ∗†</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="41716945" href="https://independent.academia.edu/ZbigniewPalmowski">Zbigniew Palmowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2011</p><p class="ds-related-work--abstract ds2-5-body-sm">Abstract. In this note we give, for a spectrally negative Lévy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero which length exceeds a certain fixed period r. The formula involves only the scale function of the spectrally negative Lévy process and the distribution of the process at time r.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Processes ∗†","attachmentId":101896059,"attachmentType":"pdf","work_url":"https://www.academia.edu/101323264/Processes_","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/101323264/Processes_"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="71443754" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/71443754/Application_to_windows_of_Dirichlet_processes">Application to windows of Dirichlet processes</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="17558976" href="https://independent.academia.edu/CristinaDiGirolami">Cristina Di Girolami</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2011</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of χ-covariation which is a generalized notion of covariation for processes with values in two Banach spaces B1 and B2. χ refers to a suitable subspace of the dual of the projective tensor product of B1 and B2. We investigate some C 1 type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If X 1 and X 2 admit a χ-covariation, F i : Bi → R, i = 1, 2 are of class C 1 with some supplementary assumptions then the covariation of the real processes F 1 (X 1) and F 2 (X 2) exist. A detailed analysis will be devoted to the so-called window processes. Let X be a real continuous process; the C([−τ, 0])-valued process X(•) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0], is called window process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for B = R. If X = X is a real valued Dirichlet process and F : B → R of class C 1 (B) then F (X) is still a Dirichlet process. If X = X is a weak Dirichlet process with finite quadratic variation, and F : C 0,1 ([0, T ] × B) is of class C 0,1 , then (F (t, Xt)) is a weak Dirichlet process. We specify corresponding results when B = C([−τ, 0]) and X = X(•). This will consitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As applications, we give a new technique for representing path-dependent random variables.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Application to windows of Dirichlet processes","attachmentId":80783918,"attachmentType":"pdf","work_url":"https://www.academia.edu/71443754/Application_to_windows_of_Dirichlet_processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/71443754/Application_to_windows_of_Dirichlet_processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":115484111,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":115484111,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_115484111" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="40464958" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/40464958/Essentials_of_Stochastic_Processes">Essentials of Stochastic Processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="128512527" href="https://duke.academia.edu/JiaqiLi">Jiaqi Li</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Essentials of Stochastic Processes","attachmentId":60730579,"attachmentType":"pdf","work_url":"https://www.academia.edu/40464958/Essentials_of_Stochastic_Processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/40464958/Essentials_of_Stochastic_Processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="1" data-entity-id="92343621" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/92343621/Theory_of_processes">Theory of processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="196367485" href="https://independent.academia.edu/AndrewMironov1">Andrew Mironov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Cornell University - arXiv, 2010</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Theory of processes","attachmentId":95377244,"attachmentType":"pdf","work_url":"https://www.academia.edu/92343621/Theory_of_processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/92343621/Theory_of_processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="2" data-entity-id="17820414" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/17820414/On_infinite_series_of_independent_Ornstein_Uhlenbeck_processes">On infinite series of independent Ornstein-Uhlenbeck processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="37768780" href="https://independent.academia.edu/EndreCs%C3%A1ki">Endre Csáki</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38008118" href="https://independent.academia.edu/PRevesz">P. Revesz</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="37741310" href="https://carleton-ca.academia.edu/Mikl%C3%B3sCs%C3%B6rg%C5%91">Miklós Csörgő</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Stochastic Processes and their Applications, 1991</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On infinite series of independent Ornstein-Uhlenbeck processes","attachmentId":42225599,"attachmentType":"pdf","work_url":"https://www.academia.edu/17820414/On_infinite_series_of_independent_Ornstein_Uhlenbeck_processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/17820414/On_infinite_series_of_independent_Ornstein_Uhlenbeck_processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="3" data-entity-id="33885604" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/33885604/A_martingale_characterization_of_the_Wiener_process">A martingale characterization of the Wiener process</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39761393" href="https://independent.academia.edu/WesolowskiJacek">Jacek Wesolowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Statistics & Probability Letters, 1990</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A martingale characterization of the Wiener process","attachmentId":53860700,"attachmentType":"pdf","work_url":"https://www.academia.edu/33885604/A_martingale_characterization_of_the_Wiener_process","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/33885604/A_martingale_characterization_of_the_Wiener_process"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="4" data-entity-id="40374383" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/40374383/Lecture_Notes_on_Probability_Theory_and_Random_Processes">Lecture Notes on Probability Theory and Random Processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="126832562" href="https://jntuhceh.academia.edu/msree">mounika sree</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Lecture Notes on Probability Theory and Random Processes","attachmentId":60622941,"attachmentType":"pdf","work_url":"https://www.academia.edu/40374383/Lecture_Notes_on_Probability_Theory_and_Random_Processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/40374383/Lecture_Notes_on_Probability_Theory_and_Random_Processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="5" data-entity-id="85677110" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/85677110/On_path_properties_of_certain_infinitely_divisible_processes">On path properties of certain infinitely divisible processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38739040" href="https://independent.academia.edu/JanRosi%C5%84ski">Jan Rosiński</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Stochastic Processes and their Applications, 1989</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On path properties of certain infinitely divisible processes","attachmentId":90302569,"attachmentType":"pdf","work_url":"https://www.academia.edu/85677110/On_path_properties_of_certain_infinitely_divisible_processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/85677110/On_path_properties_of_certain_infinitely_divisible_processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="6" data-entity-id="78427929" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/78427929/A_martingale_approach_to_some_Wiener_Hopf_problems_II">A martingale approach to some Wiener-Hopf problems, II</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38365502" href="https://swansea.academia.edu/DWilliams">David Williams</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Lecture Notes in Mathematics, 1982</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A martingale approach to some Wiener-Hopf problems, II","attachmentId":85479404,"attachmentType":"pdf","work_url":"https://www.academia.edu/78427929/A_martingale_approach_to_some_Wiener_Hopf_problems_II","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/78427929/A_martingale_approach_to_some_Wiener_Hopf_problems_II"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="7" data-entity-id="120282239" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths">Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="55567278" href="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski">Paweł J Szabłowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2022</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths","attachmentId":115484116,"attachmentType":"pdf","work_url":"https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/120282239/Stationary_Markov_stochastic_processes_with_polynomial_conditional_moments_and_continuous_paths"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="8" data-entity-id="59589888" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/59589888/Notes_on_Brownian_motion_and_related_phenomena">Notes on Brownian motion and related phenomena</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48276397" href="https://independent.academia.edu/DebShankarRay">Deb Shankar Ray</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Eprint Arxiv Physics 9903033, 1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Notes on Brownian motion and related phenomena","attachmentId":73435457,"attachmentType":"pdf","work_url":"https://www.academia.edu/59589888/Notes_on_Brownian_motion_and_related_phenomena","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/59589888/Notes_on_Brownian_motion_and_related_phenomena"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="9" data-entity-id="84632477" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/84632477/Stochastic_analysis_and_applications">Stochastic analysis and applications</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26725498" href="https://independent.academia.edu/AnaCruzeiro">Ana Cruzeiro</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Bulletin of the American Mathematical Society, 2002</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Stochastic analysis and applications","attachmentId":89586626,"attachmentType":"pdf","work_url":"https://www.academia.edu/84632477/Stochastic_analysis_and_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/84632477/Stochastic_analysis_and_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="10" data-entity-id="79039358" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/79039358/Stochastic_Processes">Stochastic Processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="12381070" href="https://uv-cl.academia.edu/RolandoRebolledo">Rolando Rebolledo</a></div><p class="ds-related-work--metadata ds2-5-body-xs">International Encyclopedia of Statistical Science, 2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Stochastic Processes","attachmentId":85893858,"attachmentType":"pdf","work_url":"https://www.academia.edu/79039358/Stochastic_Processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/79039358/Stochastic_Processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="11" data-entity-id="56031960" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/56031960/The_generalized_Wiener_process_II_Finite_systems">The generalized Wiener process II: Finite systems</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="181175070" href="https://independent.academia.edu/ManuelCaceres52">Manuel Caceres</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The generalized Wiener process II: Finite systems","attachmentId":71617812,"attachmentType":"pdf","work_url":"https://www.academia.edu/56031960/The_generalized_Wiener_process_II_Finite_systems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/56031960/The_generalized_Wiener_process_II_Finite_systems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="12" data-entity-id="27325911" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/27325911/On_the_study_of_processes_of_classes_sum_H_and_sum_s_H_">On the study of processes of classes $\sum(H)$ and $\sum_{s}(H)$</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51438452" href="https://independent.academia.edu/FulgenceObiang">Fulgence Eyi Obiang</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the study of processes of classes $\\sum(H)$ and $\\sum_{s}(H)$","attachmentId":47582835,"attachmentType":"pdf","work_url":"https://www.academia.edu/27325911/On_the_study_of_processes_of_classes_sum_H_and_sum_s_H_","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/27325911/On_the_study_of_processes_of_classes_sum_H_and_sum_s_H_"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="13" data-entity-id="10949788" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/10949788/Introduction_to_Stochastic_Processes">Introduction to Stochastic Processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26536485" href="https://independent.academia.edu/SvetlozarRachev">Svetlozar Rachev</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Handbook of Finance, 2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Introduction to Stochastic Processes","attachmentId":47004751,"attachmentType":"pdf","work_url":"https://www.academia.edu/10949788/Introduction_to_Stochastic_Processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/10949788/Introduction_to_Stochastic_Processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="14" data-entity-id="45098301" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/45098301/Applied_Stochastic_Processes">Applied Stochastic Processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="156164789" href="https://independent.academia.edu/pandaJyotirmayee">Jyotirmayee panda</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="140422877" href="https://isical.academia.edu/RamakrushnaMishra">Ramakrushna Mishra</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Applied Stochastic Processes","attachmentId":65661279,"attachmentType":"pdf","work_url":"https://www.academia.edu/45098301/Applied_Stochastic_Processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/45098301/Applied_Stochastic_Processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="15" data-entity-id="117032137" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/117032137/New_contributions_to_the_study_of_stochastic_processes_of_the_class_Sigma_">New contributions to the study of stochastic processes of the class $(\Sigma)$</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="150847270" href="https://ustmasuku.academia.edu/OctaveMoutsinga">Octave Moutsinga</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv: Probability, 2018</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"New contributions to the study of stochastic processes of the class $(\\Sigma)$","attachmentId":112999366,"attachmentType":"pdf","work_url":"https://www.academia.edu/117032137/New_contributions_to_the_study_of_stochastic_processes_of_the_class_Sigma_","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/117032137/New_contributions_to_the_study_of_stochastic_processes_of_the_class_Sigma_"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="16" data-entity-id="69539132" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/69539132/Probability_and_Stochastic_Processes_with_Applications">Probability and Stochastic Processes with Applications</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="99076177" href="https://harvard.academia.edu/OKnill">Oliver Knill</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Probability and Stochastic Processes with Applications","attachmentId":79596212,"attachmentType":"pdf","work_url":"https://www.academia.edu/69539132/Probability_and_Stochastic_Processes_with_Applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/69539132/Probability_and_Stochastic_Processes_with_Applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="17" data-entity-id="123482221" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/123482221/A_Generalization_of_the_Ornstein_Uhlenbeck_Process_Theoretical_Results_Simulations_and_Parameter_Estimation">A Generalization of the Ornstein-Uhlenbeck Process: Theoretical Results, Simulations and Parameter Estimation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="312528098" href="https://independent.academia.edu/AryVasconcelosMedino">Ary Vasconcelos Medino</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2021</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Generalization of the Ornstein-Uhlenbeck Process: Theoretical Results, Simulations and Parameter Estimation","attachmentId":117903080,"attachmentType":"pdf","work_url":"https://www.academia.edu/123482221/A_Generalization_of_the_Ornstein_Uhlenbeck_Process_Theoretical_Results_Simulations_and_Parameter_Estimation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/123482221/A_Generalization_of_the_Ornstein_Uhlenbeck_Process_Theoretical_Results_Simulations_and_Parameter_Estimation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="18" data-entity-id="16836882" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16836882/Individuating_processes">Individuating processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="6017617" href="https://durham.academia.edu/JohnPemberton">John Pemberton</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Individuation, Process and Scientific Practices (OUP) , 2018</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Individuating processes","attachmentId":54470483,"attachmentType":"pdf","work_url":"https://www.academia.edu/16836882/Individuating_processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/16836882/Individuating_processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="19" data-entity-id="26187658" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/26187658/On_the_small_deviation_problem_for_some_iterated_processes">On the small deviation problem for some iterated processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="50088396" href="https://independent.academia.edu/MikhailLifshits">Mikhail Lifshits</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Electronic Journal of Probability, 2009</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the small deviation problem for some iterated processes","attachmentId":46511140,"attachmentType":"pdf","work_url":"https://www.academia.edu/26187658/On_the_small_deviation_problem_for_some_iterated_processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/26187658/On_the_small_deviation_problem_for_some_iterated_processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="20" data-entity-id="96030241" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/96030241/On_a_class_of_Markov_processes_in_Hilbert_space">On a class of Markov processes in Hilbert space</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="59989690" href="https://independent.academia.edu/RichardDuncan13">Richard Duncan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematische Zeitschrift, 1973</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On a class of Markov processes in Hilbert space","attachmentId":98042756,"attachmentType":"pdf","work_url":"https://www.academia.edu/96030241/On_a_class_of_Markov_processes_in_Hilbert_space","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/96030241/On_a_class_of_Markov_processes_in_Hilbert_space"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="21" data-entity-id="69539130" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/69539130/Probability_Theory_and_Stochastic_Processes_with_Applications">Probability Theory and Stochastic Processes with Applications</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="99076177" href="https://harvard.academia.edu/OKnill">Oliver Knill</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Probability Theory and Stochastic Processes with Applications","attachmentId":79596259,"attachmentType":"pdf","work_url":"https://www.academia.edu/69539130/Probability_Theory_and_Stochastic_Processes_with_Applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/69539130/Probability_Theory_and_Stochastic_Processes_with_Applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="22" data-entity-id="65128648" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/65128648/Sample_Path_Behaviour_of_Wiener_and_Cauchy_Processes">Sample Path Behaviour of Wiener and Cauchy Processes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="43055484" href="https://independent.academia.edu/StanislavKeprta">Stanislav Keprta</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1998</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Sample Path Behaviour of Wiener and Cauchy Processes","attachmentId":76855203,"attachmentType":"pdf","work_url":"https://www.academia.edu/65128648/Sample_Path_Behaviour_of_Wiener_and_Cauchy_Processes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/65128648/Sample_Path_Behaviour_of_Wiener_and_Cauchy_Processes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="23" data-entity-id="91006767" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/91006767/Smooth_stationary_stochastic_processes_with_polynomial_conditional_moments">Smooth stationary stochastic processes with polynomial conditional moments</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="55567278" href="https://racjonalista.academia.edu/Pawe%C5%82Szab%C5%82owski">Paweł J Szabłowski</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2022</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Smooth stationary stochastic processes with polynomial conditional moments","attachmentId":94471806,"attachmentType":"pdf","work_url":"https://www.academia.edu/91006767/Smooth_stationary_stochastic_processes_with_polynomial_conditional_moments","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/91006767/Smooth_stationary_stochastic_processes_with_polynomial_conditional_moments"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="24" data-entity-id="55034332" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/55034332/Statistical_problems_of_the_elementary_Gaussian_processes_II_Statistics_and_related_problems">Statistical problems of the elementary Gaussian processes: II. Statistics and related problems</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="56219195" href="https://independent.academia.edu/Andr%C3%A1sBencz%C3%BAr">András Benczúr</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Statistical problems of the elementary Gaussian processes: II. Statistics and related problems","attachmentId":71102841,"attachmentType":"pdf","work_url":"https://www.academia.edu/55034332/Statistical_problems_of_the_elementary_Gaussian_processes_II_Statistics_and_related_problems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/55034332/Statistical_problems_of_the_elementary_Gaussian_processes_II_Statistics_and_related_problems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related topics</h2><div class="ds-research-interests--pills-container"><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="300" rel="nofollow" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="54284" rel="nofollow" href="https://www.academia.edu/Documents/in/Generalization">Generalization</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="389829" rel="nofollow" href="https://www.academia.edu/Documents/in/Gaussian_distribution">Gaussian distribution</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="679783" rel="nofollow" href="https://www.academia.edu/Documents/in/Boolean_Satisfiability">Boolean Satisfiability</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="741671" rel="nofollow" href="https://www.academia.edu/Documents/in/Markov_Process">Markov Process</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="2382100" rel="nofollow" href="https://www.academia.edu/Documents/in/Correlation_function">Correlation function</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="3115633" rel="nofollow" href="https://www.academia.edu/Documents/in/Discrete_time">Discrete time</a></div></div></div></div></div><div class="footer--content"><ul class="footer--main-links hide-on-mobile"><li><a href="https://www.academia.edu/about">About</a></li><li><a href="https://www.academia.edu/press">Press</a></li><li><a href="https://www.academia.edu/documents">Papers</a></li><li><a href="https://www.academia.edu/topics">Topics</a></li><li><a href="https://www.academia.edu/hiring"><svg style="width: 13px; height: 13px; position: relative; bottom: -1px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg> <strong>We're Hiring!</strong></a></li><li><a href="https://support.academia.edu/hc/en-us"><svg style="width: 12px; height: 12px; position: relative; bottom: -1px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg> <strong>Help Center</strong></a></li></ul><ul class="footer--research-interests"><li>Find new research papers in:</li><li><a href="https://www.academia.edu/Documents/in/Physics">Physics</a></li><li><a href="https://www.academia.edu/Documents/in/Chemistry">Chemistry</a></li><li><a href="https://www.academia.edu/Documents/in/Biology">Biology</a></li><li><a href="https://www.academia.edu/Documents/in/Health_Sciences">Health Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Ecology">Ecology</a></li><li><a href="https://www.academia.edu/Documents/in/Earth_Sciences">Earth Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Cognitive_Science">Cognitive Science</a></li><li><a href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></li><li><a href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a></li></ul><ul class="footer--legal-links hide-on-mobile"><li><a href="https://www.academia.edu/terms">Terms</a></li><li><a href="https://www.academia.edu/privacy">Privacy</a></li><li><a href="https://www.academia.edu/copyright">Copyright</a></li><li>Academia ©2025</li></ul></div> </body> </html>