exercise:25029bf420: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> (Alternate proof of Theorem) Let <math>\mat{P}</math> be the transition matrix of an ergodic Markov chain. Let <math>\mat{x}</math> be any column vector such that <math>\mat{P} \mat{x} = \mat{ x}</math>. Let <m...") |
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\newcommand{\secstoprocess}{\all} | \newcommand{\secstoprocess}{\all} | ||
\newcommand{\NA}{{\rm NA}} | \newcommand{\NA}{{\rm NA}} | ||
\newcommand{\mathds}{\mathbb}</math></div> (Alternate proof of [[guide:E5c38a1e8a#thm 11.3.8 |Theorem]]) Let | \newcommand{\mathds}{\mathbb}</math></div> (Alternate proof of [[guide:E5c38a1e8a#thm 11.3.8 |Theorem]]) Let <math>\mat{P}</math> be the transition matrix of an ergodic Markov chain. Let <math>\mat{x}</math> be any column vector such that | ||
<math>\mat{P}</math> be the | |||
transition matrix of an ergodic Markov chain. Let <math>\mat{x}</math> be any column | |||
vector such that | |||
<math>\mat{P} | <math>\mat{P} | ||
\mat{x} = | \mat{x} = | ||
\mat{ x}</math>. Let <math>M</math> be the maximum value of the components of <math>\mat{x}</math>. | \mat{ x}</math>. Let <math>M</math> be the maximum value of the components of <math>\mat{x}</math>. | ||
Assume that <math>x_i | Assume that <math>x_i = M</math>. Show that if <math>p_{ij} > 0</math> then <math>x_j = M</math>. Use this to prove that | ||
= M</math>. Show that if <math>p_{ij} > 0</math> then <math>x_j = M</math>. Use this to prove that | <math>\mat{x}</math> must be a constant vector. | ||
<math>\mat{x}</math> | |||
must be a constant vector. | |||
Latest revision as of 23:07, 17 June 2024
[math]
\newcommand{\NA}{{\rm NA}}
\newcommand{\mat}[1]{{\bf#1}}
\newcommand{\exref}[1]{\ref{##1}}
\newcommand{\secstoprocess}{\all}
\newcommand{\NA}{{\rm NA}}
\newcommand{\mathds}{\mathbb}[/math]
(Alternate proof of Theorem) Let [math]\mat{P}[/math] be the transition matrix of an ergodic Markov chain. Let [math]\mat{x}[/math] be any column vector such that
[math]\mat{P} \mat{x} = \mat{ x}[/math]. Let [math]M[/math] be the maximum value of the components of [math]\mat{x}[/math]. Assume that [math]x_i = M[/math]. Show that if [math]p_{ij} \gt 0[/math] then [math]x_j = M[/math]. Use this to prove that [math]\mat{x}[/math] must be a constant vector.