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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Long-range_dependency&redirect=no" class="mw-redirect" title="Long-range dependency">Long-range dependency</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Phenomenon in linguistics and data analysis</div> <p><b>Long-range dependence</b> (<b>LRD</b>), also called <b>long memory</b> or <b>long-range persistence</b>, is a phenomenon that may arise in the analysis of <a href="/wiki/Spatial_analysis" title="Spatial analysis">spatial</a> or <a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">time series</a> data. It relates to the rate of decay of <a href="/wiki/Statistical_dependence" class="mw-redirect" title="Statistical dependence">statistical dependence</a> of two points with increasing time interval or spatial distance between the points. A phenomenon is usually considered to have long-range dependence if the dependence decays more slowly than an <a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a>, typically a power-like decay. LRD is often related to <a href="/wiki/Self-similar_process" title="Self-similar process">self-similar processes</a> or fields. LRD has been used in various fields such as internet traffic modelling, <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, <a href="/wiki/Hydrology" title="Hydrology">hydrology</a>, <a href="/wiki/Linguistics" title="Linguistics">linguistics</a> and the earth sciences. Different mathematical definitions of LRD are used for different contexts and purposes.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Short-range_dependence_versus_long-range_dependence">Short-range dependence versus long-range dependence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Long-range_dependence&action=edit&section=1" title="Edit section: Short-range dependence versus long-range dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One way of characterising long-range and short-range dependent stationary process is in terms of their <a href="/wiki/Autocovariance" title="Autocovariance">autocovariance</a> functions. For a short-range dependent process, the coupling between values at different times decreases rapidly as the time difference increases. Either the autocovariance drops to zero after a certain time-lag, or it eventually has an <a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a>. In the case of LRD, there is much stronger coupling. The decay of the autocovariance function is power-like and so is slower than exponential. </p><p>A second way of characterizing long- and short-range dependence is in terms of the variance of partial sum of consecutive values. For short-range dependence, the variance grows typically proportionally to the number of terms. As for LRD, the variance of the partial sum increases more rapidly which is often a power function with the exponent greater than 1. A way of examining this behavior uses the <a href="/wiki/Rescaled_range" title="Rescaled range">rescaled range</a>. This aspect of long-range dependence is important in the design of <a href="/wiki/Dam" title="Dam">dams</a> on rivers for <a href="/wiki/Water_resources" title="Water resources">water resources</a>, where the summations correspond to the total inflow to the dam over an extended period.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The above two ways are mathematically related to each other, but they are not the only ways to define LRD. In the case where the autocovariance of the process does not exist (<a href="/wiki/Heavy_tails" class="mw-redirect" title="Heavy tails">heavy tails</a>), one has to find other ways to define what LRD means, and this is often done with the help of <a href="/wiki/Self-similar_process" title="Self-similar process">self-similar processes</a>. </p><p>The <a href="/wiki/Hurst_parameter" class="mw-redirect" title="Hurst parameter">Hurst parameter</a> <i>H</i> is a measure of the extent of long-range dependence in a time series (while it has another meaning in the context of <a href="/wiki/Self-similar_process" title="Self-similar process">self-similar processes</a>). <i>H</i> takes on values from 0 to 1. A value of 0.5 indicates the absence of long-range dependence.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The closer <i>H</i> is to 1, the greater the degree of persistence or long-range dependence. <i>H</i> less than 0.5 corresponds to anti-persistency, which as the opposite of LRD indicates strong negative correlation so that the process fluctuates violently. </p> <div class="mw-heading mw-heading2"><h2 id="Estimation_of_the_Hurst_parameter">Estimation of the Hurst parameter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Long-range_dependence&action=edit&section=2" title="Edit section: Estimation of the Hurst parameter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Slowly decaying variances, LRD, and a spectral density obeying a power-law are different manifestations of the property of the underlying covariance of a stationary process. Therefore, it is possible to approach the problem of estimating the Hurst parameter from three difference angles: </p> <ul><li>Variance-time plot: based on the analysis of the variances of the aggregate processes</li> <li>R/S statistics: based on the time-domain analysis of the rescaled adjusted range</li> <li>Periodogram: based on a frequency-domain analysis</li></ul> <div class="mw-heading mw-heading2"><h2 id="Relation_to_self-similar_processes">Relation to self-similar processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Long-range_dependence&action=edit&section=3" title="Edit section: Relation to self-similar processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a stationary LRD sequence, the partial sum if viewed as a process indexed by the number of terms after a proper scaling, is a <a href="/wiki/Self-similar_process" title="Self-similar process">self-similar process</a> with <a href="/wiki/Stationary_increments" title="Stationary increments">stationary increments</a> asymptotically, the most typical one being <a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">fractional Brownian motion</a>. In the converse, given a self-similar process with stationary increments with Hurst index <i>H</i> > 0.5, its increments (consecutive differences of the process) is a stationary LRD sequence. </p><p>This also holds true if the sequence is short-range dependent, but in this case the self-similar process resulting from the partial sum can only be <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> (<i>H</i> = 0.5). </p> <div class="mw-heading mw-heading2"><h2 id="Models">Models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Long-range_dependence&action=edit&section=4" title="Edit section: Models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Among <a href="/wiki/Stochastic_model" class="mw-redirect" title="Stochastic model">stochastic models</a> that are used for long-range dependence, some popular ones are <a href="/wiki/Autoregressive_fractionally_integrated_moving_average" title="Autoregressive fractionally integrated moving average">autoregressive fractionally integrated moving average</a> models, which are defined for discrete-time processes, while continuous-time models might start from <a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">fractional Brownian motion</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Long-range_dependence&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Long-tail_traffic" title="Long-tail traffic">Long-tail traffic</a></li> <li><a href="/wiki/Traffic_generation_model" title="Traffic generation model">Traffic generation model</a></li> <li><a href="/wiki/Detrended_fluctuation_analysis" title="Detrended fluctuation analysis">Detrended fluctuation analysis</a></li> <li><a href="/wiki/Tweedie_distributions" class="mw-redirect" title="Tweedie distributions">Tweedie distributions</a></li> <li><a href="/wiki/Fractal_dimension" title="Fractal dimension">Fractal dimension</a></li> <li><a href="/wiki/Hurst_exponent" title="Hurst exponent">Hurst exponent</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Long-range_dependence&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output 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a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBeran1994" class="citation book cs1">Beran, Jan (1994). <i>Statistics for Long-Memory Processes</i>. CRC Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+for+Long-Memory+Processes&rft.pub=CRC+Press&rft.date=1994&rft.aulast=Beran&rft.aufirst=Jan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoukhan2003" class="citation book cs1">Doukhan; et al. (2003). <i>Theory and Applications of Long-Range Dependence</i>. Birkhäuser.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+and+Applications+of+Long-Range+Dependence&rft.pub=Birkh%C3%A4user&rft.date=2003&rft.au=Doukhan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMalamudTurcotte1999" class="citation book cs1">Malamud, Bruce D.; Turcotte, Donald L. (1999). <i>Self-Affine Time Series: I. Generation and Analyses</i>. Vol. 40. pp. <span class="nowrap">1–</span>90. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999AdGeo..40....1M">1999AdGeo..40....1M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0065-2687%2808%2960293-9">10.1016/S0065-2687(08)60293-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780120188406" title="Special:BookSources/9780120188406"><bdi>9780120188406</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Self-Affine+Time+Series%3A+I.+Generation+and+Analyses&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E90&rft.date=1999&rft_id=info%3Adoi%2F10.1016%2FS0065-2687%2808%2960293-9&rft_id=info%3Abibcode%2F1999AdGeo..40....1M&rft.isbn=9780120188406&rft.aulast=Malamud&rft.aufirst=Bruce+D.&rft.au=Turcotte%2C+Donald+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|journal=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamorodnitsky2007" class="citation book cs1">Samorodnitsky, Gennady (2007). <i>Long range dependence</i>. Foundations and Trends in Stochastic Systems.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Long+range+dependence&rft.pub=Foundations+and+Trends+in+Stochastic+Systems&rft.date=2007&rft.aulast=Samorodnitsky&rft.aufirst=Gennady&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeran2013" class="citation book cs1">Beran; et al. (2013). <i>Long memory processes: probabilistic properties and statistical methods</i>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Long+memory+processes%3A+probabilistic+properties+and+statistical+methods&rft.pub=Springer&rft.date=2013&rft.au=Beran&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWittMalamud2013" class="citation journal cs1">Witt, Annette; Malamud, Bruce D. (September 2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10712-012-9217-8">"Quantification of Long-Range Persistence in Geophysical Time Series: Conventional and Benchmark-Based Improvement Techniques"</a>. <i>Surveys in Geophysics</i>. <b>34</b> (5): <span class="nowrap">541–</span>651. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013SGeo...34..541W">2013SGeo...34..541W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10712-012-9217-8">10.1007/s10712-012-9217-8</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Surveys+in+Geophysics&rft.atitle=Quantification+of+Long-Range+Persistence+in+Geophysical+Time+Series%3A+Conventional+and+Benchmark-Based+Improvement+Techniques&rft.volume=34&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E541-%3C%2Fspan%3E651&rft.date=2013-09&rft_id=info%3Adoi%2F10.1007%2Fs10712-012-9217-8&rft_id=info%3Abibcode%2F2013SGeo...34..541W&rft.aulast=Witt&rft.aufirst=Annette&rft.au=Malamud%2C+Bruce+D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs10712-012-9217-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">*Hurst, H.E., Black, R.P., Simaika, Y.M. (1965) <i>Long-term storage: an experimental study</i> Constable, London.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Beran (1994) page 34</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Long-range_dependence&action=edit&section=7" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBariviera2011" class="citation journal cs1">Bariviera, A.F. (2011). "The influence of liquidity on informational efficiency: The case of the Thai Stock Market". <i>Physica A: Statistical Mechanics and Its Applications</i>. <b>390</b> (23): <span class="nowrap">4426–</span>4432. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011PhyA..390.4426B">2011PhyA..390.4426B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physa.2011.07.032">10.1016/j.physa.2011.07.032</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120377241">120377241</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+A%3A+Statistical+Mechanics+and+Its+Applications&rft.atitle=The+influence+of+liquidity+on+informational+efficiency%3A+The+case+of+the+Thai+Stock+Market&rft.volume=390&rft.issue=23&rft.pages=%3Cspan+class%3D%22nowrap%22%3E4426-%3C%2Fspan%3E4432&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120377241%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.physa.2011.07.032&rft_id=info%3Abibcode%2F2011PhyA..390.4426B&rft.aulast=Bariviera&rft.aufirst=A.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarivieraGuercioMartinez2012" class="citation journal cs1">Bariviera, A.F.; Guercio, M.B.; Martinez, L.B. (2012). "A comparative analysis of the informational efficiency of the fixed income market in seven European countries". <i>Economics Letters</i>. <b>116</b> (3): <span class="nowrap">426–</span>428. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.econlet.2012.04.047">10.1016/j.econlet.2012.04.047</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11336%2F66311">11336/66311</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:153323583">153323583</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Economics+Letters&rft.atitle=A+comparative+analysis+of+the+informational+efficiency+of+the+fixed+income+market+in+seven+European+countries&rft.volume=116&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E426-%3C%2Fspan%3E428&rft.date=2012&rft_id=info%3Ahdl%2F11336%2F66311&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A153323583%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.econlet.2012.04.047&rft.aulast=Bariviera&rft.aufirst=A.F.&rft.au=Guercio%2C+M.B.&rft.au=Martinez%2C+L.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrockwell2006" class="citation journal cs1">Brockwell, A.E. 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"Likelihood-based analysis of a class of generalized long-memory time series models". <i>Journal of Time Series Analysis</i>. <b>28</b> (3): <span class="nowrap">386–</span>407. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1467-9892.2006.00515.x">10.1111/j.1467-9892.2006.00515.x</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122206112">122206112</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Time+Series+Analysis&rft.atitle=Likelihood-based+analysis+of+a+class+of+generalized+long-memory+time+series+models&rft.volume=28&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E386-%3C%2Fspan%3E407&rft.date=2006&rft_id=info%3Adoi%2F10.1111%2Fj.1467-9892.2006.00515.x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122206112%23id-name%3DS2CID&rft.aulast=Brockwell&rft.aufirst=A.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALong-range+dependence" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGranger,_C._W._J.Joyeux,_R.1980" class="citation journal cs1"><a href="/w/index.php?title=Granger,_C.&action=edit&redlink=1" class="new" title="Granger, C. 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