{"id":6422,"date":"2018-06-20T09:45:21","date_gmt":"2018-06-20T08:45:21","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=6422"},"modified":"2022-12-03T23:00:45","modified_gmt":"2022-12-03T22:00:45","slug":"loi-invariante-et-comportement-asymptotique","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/invariant-law-and-asymptotic-behavior\/","title":{"rendered":"Invariant law and asymptotic behavior"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"6422\" class=\"elementor elementor-6422\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-953d150 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"953d150\" 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 elementor-top-column elementor-element elementor-element-40b47ef\" data-id=\"40b47ef\" 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-ae7a759 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"ae7a759\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Markov process<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/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<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-61ca07a\" data-id=\"61ca07a\" 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-35625b8 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"35625b8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Home page<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/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<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-20149a9\" data-id=\"20149a9\" 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-552b033 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"552b033\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/fr.wikipedia.org\/wiki\/Cha%C3%AEne_de_Markov\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-70e26bd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"70e26bd\" 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-36a76ec\" data-id=\"36a76ec\" 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-ee16a47 elementor-widget elementor-widget-progress\" data-id=\"ee16a47\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"progress.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<span class=\"elementor-title\" id=\"elementor-progress-bar-ee16a47\">\n\t\t\t\tDifficulty\t\t\t<\/span>\n\t\t\n\t\t<div aria-labelledby=\"elementor-progress-bar-ee16a47\" class=\"elementor-progress-wrapper\" role=\"progressbar\" aria-valuemin=\"0\" aria-valuemax=\"100\" aria-valuenow=\"50\" aria-valuetext=\"50% (Moyen)\">\n\t\t\t<div class=\"elementor-progress-bar\" data-max=\"50\">\n\t\t\t\t<span class=\"elementor-progress-text\">Average<\/span>\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-progress-percentage\">50%<\/span>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\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<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3edadbf8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3edadbf8\" 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-7c774bef\" data-id=\"7c774bef\" 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-4fa178e4 elementor-widget elementor-widget-text-editor\" data-id=\"4fa178e4\" 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_85 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\/invariant-law-and-asymptotic-behavior\/#Loi-invariante-et-comportement-asymptotique\" >Invariant law and asymptotic behavior<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/invariant-law-and-asymptotic-behavior\/#Idee\" >Idea<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/invariant-law-and-asymptotic-behavior\/#Periodicite\" >Periodicity<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/invariant-law-and-asymptotic-behavior\/#Loi-invariante\" >Invariant law<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/invariant-law-and-asymptotic-behavior\/#Calcul-exacte-de-la-mesure\" >Exact calculation of the measurement<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/invariant-law-and-asymptotic-behavior\/#Calcul-exacte-de-la-mesure-cas-non-ergodique\" >Exact calculation of the measurement (non-ergodic case)<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Loi-invariante-et-comportement-asymptotique\"><\/span>Invariant law and asymptotic behavior<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"wp-block-paragraph\">We seek to understand the asymptotic behavior of a homogeneous Markov chain. That is to say the limit of the transition probabilities Q<sup>not<\/sup>(i, j) when n becomes very large (the invariant law).<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Idee\"><\/span>Idea<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"wp-block-paragraph\">We seek to answer the following question: &quot;What is the probability that after n<br \/>not here <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chains\/\">markov chain<\/a> be in a given state? &quot;. Take the following transition matrix P:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" class=\"alignnone wp-image-6570\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba9.png\" alt=\"invariant law\" width=\"128\" height=\"81\" title=\"\"><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">Suppose none of the machines are down on the first day. Then we have as initial vector (1, 0), to calculate the following distribution we multiply the initial vector by P, etc. Which gives the following results:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6571\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba10.png\" alt=\"invariant law\" width=\"397\" height=\"197\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba10.png 397w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba10-300x149.png 300w\" sizes=\"(max-width: 397px) 100vw, 397px\" \/><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">This means that after 10 iteration, if we consider that we are starting from state 0, we have 99% to be in state 0 and 1% in state 1. This can also be interpreted as follows: on a starting population, considering the initial distribution by vector (1.0), then 99% of the population will be in state 0 and 1% in state 1.<\/p>\n\n<p class=\"wp-block-paragraph\">We notice that the fourth and the tenth iteration have similar results (here identical to 4 significant digits). We then speak of a law of convergence, and this law does not depend on the distribution at the origin.<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Periodicite\"><\/span>Periodicity<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"wp-block-paragraph\">Let us take the example of the following matrix P = {0, 1; 1, 0}. We note that P\u00b2 = Id which implies the following relation: \u2200n\u2208\u039d, P<sup>2n + 1<\/sup>= P.<\/p>\n\n<p class=\"wp-block-paragraph\">Such a chain does not converge, we say that it is 2-periodic or of period 2.<\/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 chain is said to be k-periodic iff: \u2200 (n, k) \u2208\u039d\u00b2, P<sup>kn + 1<\/sup>= P. The states of a class all have the same period.<\/div>\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 state x is said to be aperiodic if P<sup>not<\/sup>(x, x)&gt; 0. If P is irreducible and has at least one aperiodic state, then all the states are aperiodic.<\/div>\n\n<p class=\"wp-block-paragraph\">Before calculating the invariant law of a Markov chain, it is necessary to verify that the latter is irreducible and aperiodic (also called ergodic).<\/p>\n\n<p class=\"wp-block-paragraph\">A chain is therefore ergodic if any state can be reached from any other state and for a power P<sup>k<\/sup> all elements are strictly positive. It is therefore possible to go from one state to another in at most k stages, independently of the starting and ending points. An ergodic chain has an invariant law (be careful, it is also possible to calculate the stationary distribution of the other chains, the interpretation is not the same).<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Loi-invariante\"><\/span>Invariant law<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"wp-block-paragraph\">One says that a probability measure \u03c0 is invariant or stationary if for a transition matrix P we have \u03c0P = \u03c0. Note that since \u03c0 is a measure then the sum of these terms is equal to 1.<\/p>\n\n<p class=\"wp-block-paragraph\">Let (Xn) defining a homogeneous Markov chain with P an irreducible aperiodic transition matrix having an invariant measure \u03c0. Then :<\/p>\n\n<ul class=\"wp-block-list\">\n<li>P (Xn = x) \u2192 \u03c0 (x) when n \u2192 \u221e for all x<\/li>\n<li>p<sup>not<\/sup>(x, y) \u2192 \u03c0 (y) when n \u2192 \u221e for all x, y<\/li>\n<\/ul>\n\n<p class=\"wp-block-paragraph\">The speed of convergence towards the stationary law is of the order of | \u03b6 |<sup>not<\/sup>\u00a0where \u03b6 is the eigenvalue of P different from 1 and of greater modulus (which is strictly less than 1).<\/p>\n\n<p class=\"wp-block-paragraph\">If the chain is ergodic (irreducible and aperiodic) then all states are reachable from any other state. Such a chain has an invariant law.<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Calcul-exacte-de-la-mesure\"><\/span>Exact calculation of the measurement<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"wp-block-paragraph\">Let us take the definition \u00b5P = \u00b5 knowing that \u00b5 is stochastic (and yes I changed the name of the initial distribution!). This gives the following linear system:<\/p>\n\n<div style=\"padding: 5px; background-color: #ffdcd3; border: 2px solid #ff7964; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">\n<figure><img decoding=\"async\" class=\"aligncenter wp-image-6423 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/06\/markov11.png\" alt=\"invariant law\" width=\"233\" height=\"74\" title=\"\"><\/figure>\n<p>A measure \u00b5 on E is invariant (for P) if \u00b5 = \u00b5P, i.e. for all y \u2208 E<\/p>\n<p>We speak of an invariant law if in addition \u00b5 is a probability (\u00b5 (E) = 1). We also say invariant \/ stationary law \/ probability.<\/p>\n<\/div>\n\n<p class=\"wp-block-paragraph\">Given the relation \u00b5<sub>n + 1<\/sub> = \u00b5<sub>not<\/sub>P, we see immediately that, if \u00b5 is an invariant law and X<sub>0<\/sub> \u223c \u00b5, then X<sub>not<\/sub> \u223c \u00b5 for all n. We also notice that \u00b5 does not depend on the initial distribution vector.<\/p>\n\n<p class=\"wp-block-paragraph\">Take for example a three-state Markov chain, whose transition matrix is as follows:<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6572\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba11.png\" alt=\"invariant law\" width=\"145\" height=\"69\" title=\"\"><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">We notice immediately that the matrix forms an irreducible and aperiodic chain, since all the states communicate and that p<sub>ii<\/sub>\u00a0&gt; 0. We seek to solve the system \u00b5P = \u00b5 with for solution \u00b5 * = (p, q, r) such that p + q + r = 1 and 0 &lt;p,q,r &lt;1 ce qui donne les \u00e9quations suivantes :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6573\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba12.png\" alt=\"invariant law\" width=\"227\" height=\"88\" title=\"\"><\/figure>\n<\/div>\n\n<p class=\"wp-block-paragraph\">We find as a solution the vector \u00b5 * = (2\/53, 10\/53, 41\/53).<\/p>\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Calcul-exacte-de-la-mesure-cas-non-ergodique\"><\/span>Exact calculation of the measurement (non-ergodic case)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"wp-block-paragraph\">Consider the following Markov chain<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6574\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba13.png\" alt=\"invariant law\" width=\"579\" height=\"341\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba13.png 579w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba13-300x177.png 300w\" sizes=\"(max-width: 579px) 100vw, 579px\" \/><\/figure>\n\n<p class=\"wp-block-paragraph\">We find that state 1 forms a <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/recurrence-and-transition-criteria\/\">transitional class<\/a>, state 2 forms an absorbing class and states 3, 4 form a recurrent class. Let us make the analysis of the asymptotic behavior by not taking account of its nonergotic character (no guarantee of convergence).<\/p>\n\n<p class=\"wp-block-paragraph\">Let&#039;s solve the following system:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6575\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba14.png\" alt=\"invariant law\" width=\"426\" height=\"160\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba14.png 426w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba14-300x113.png 300w\" sizes=\"(max-width: 426px) 100vw, 426px\" \/><\/figure>\n\n<p class=\"wp-block-paragraph\">The system does not allow a single solution. Let \u03b1 be between 0 and 1, then we find a solution by admitting that \u03b1 is solution of \u03c02:<\/p>\n\n<figure class=\"wp-block-image\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6576\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/09\/proba15.png\" alt=\"invariant law\" width=\"166\" height=\"140\" title=\"\"><\/figure>\n\n<p class=\"wp-block-paragraph\">We notice in the class {3,4} that the probabilities are equal on a frequency of 2k, we deduce that the class is periodic over a period of 2 (which we could have calculated by power of the matrix). Class {2} is absorbing, there is no period. The class {1} is transient, which is why there is no longer a population after a time k.<\/p>\n\n<p class=\"wp-block-paragraph\">In the case of a non-periodic chain, it is possible to deduce from the asymptotic behavior the classes of the states as well as the periodicities of the classes and of the chain.<\/p>\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 Home page Wiki Difficulty Medium 50% Invariant law and asymptotic behavior We seek to understand the asymptotic behavior of a Markov chain\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-6422","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6422","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=6422"}],"version-history":[{"count":9,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6422\/revisions"}],"predecessor-version":[{"id":18623,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/6422\/revisions\/18623"}],"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=6422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}