{"id":6578,"date":"2018-09-05T13:33:18","date_gmt":"2018-09-05T12:33:18","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=6578"},"modified":"2022-12-03T23:01:57","modified_gmt":"2022-12-03T22:01:57","slug":"criteres-de-recurrence-et-transience","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/recurrence-and-transition-criteria\/","title":{"rendered":"Recurrence and transience criteria"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"6578\" class=\"elementor elementor-6578\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fa76ada elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fa76ada\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 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elementor-element elementor-element-5d5b573c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5d5b573c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6571bd84\" data-id=\"6571bd84\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1361bcd elementor-widget elementor-widget-text-editor\" data-id=\"1361bcd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/recurrence-and-transition-criteria\/#Criteres-de-recurrence-et-transience\" >Recurrence and transience criteria<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Criteres-de-recurrence-et-transience\"><\/span>Recurrence and transience criteria<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>We will study a second classification of states depending on the type of behavior of the chain (the criteria of recurrence and transience).<\/p>\n\n<p>Let x be a state of the chain, we denote the time to reach x, denoted by T<sub>x<\/sub>, the first instant when x is visited after departure. by convention, the reach time is infinite if we never reach x. The formula is as follows (we will use the classical notations for the probabilities):<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6579\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba16.png\" alt=\"recurrence and transience\" width=\"462\" height=\"60\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba16.png 462w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba16-300x39.png 300w\" sizes=\"(max-width: 462px) 100vw, 462px\" \/><\/figure>\n<\/div>\n\n<p>If the chain starts from state x, we use the term return time.<\/p>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">\n<p>A state x is said to be recurrent if:<img decoding=\"async\" class=\"alignnone wp-image-6580 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba17.png\" alt=\"recurrence and transience\" width=\"127\" height=\"29\" title=\"\"><\/p>\n<p>State x is said to be transient or transient otherwise, i.e. when:<\/p>\n<figure><img decoding=\"async\" class=\"alignnone wp-image-6581 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba18.png\" alt=\"recurrence and transience\" width=\"263\" height=\"24\" title=\"\"><\/figure>\n<p>\u00a0<\/p>\n<\/div>\n\n<p>A state is recurrent if we are sure to return to it, it is transient if there is a non-zero probability of never coming back to it, and therefore of leaving it definitively.<\/p>\n\n<p>An equivalence class is said to be recurrent, respectively transient, if one of its vertices is recurrent, resp. transient.<\/p>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">A recurring class is closed, in other words, the probability of exiting a recurring class is zero.<\/div>\n\n<p>Let x be any state belonging to the recurrence class C. Suppose<br \/>that there exists y \u2209 C such that x \u2192 y and let us show that we have a contradiction. Note first that y does not lead to any vertex of C, because otherwise we would have y \u2192 x and therefore x \u2194 y and y \u2208 C. Moreover, we have:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6582\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba19.png\" alt=\"recurrence and transience\" width=\"218\" height=\"24\" title=\"\"><\/figure>\n\n<p>However, the probability of not going back to x is bounded inferiorly by the probability of going to y in finite time (since y does not lead to any state of C). So, we have the following relation:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6583\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba20.png\" alt=\"recurrence and transience\" width=\"202\" height=\"33\" title=\"\"><\/figure>\n\n<p>Which is a contradiction with recurrent x. We see that a recurring class is closed, but the converse is false in general, it is still verified if this class has a finite cardinality.<\/p>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">It is important to retain the following corollary: <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chains\/\">markov chain<\/a> defined on a finite state space admits at least one recurrent state.<\/div>\n\n<div>\u00a0<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Markov Process Wiki Home Difficulty Easy 25% Criteria for recurrence and transience We will study a second classification of states depending on the type \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":5007,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-6578","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6578","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=6578"}],"version-history":[{"count":6,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6578\/revisions"}],"predecessor-version":[{"id":18654,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6578\/revisions\/18654"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5007"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=6578"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}