exercise:D8cb650f0a: 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> Let <math>\mat{P}</math> be the transition matrix of an ergodic Markov chain. Let <math>\mat{w}</math> be a fixed probability vector (i.e., <math>\mat{w}</math> is a row vector with <math>\mat {w}\mat {P} = \mat {w}</math>). Show that if <math...") |
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\newcommand{\secstoprocess}{\all} | \newcommand{\secstoprocess}{\all} | ||
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\newcommand{\mathds}{\mathbb}</math></div> Let <math>\mat{P}</math> be the transition matrix of an ergodic | \newcommand{\mathds}{\mathbb}</math></div> Let <math>\mat{P}</math> be the transition matrix of an ergodic Markov chain. Let <math>\mat{w}</math> be a fixed probability vector (i.e., <math>\mat{w}</math> is a row vector with <math>\mat {w}\mat {P} = \mat {w}</math>). Show that if <math>w_i = 0</math> and <math>p_{ji} > 0</math> then <math>w_j = 0</math>. Use this to show that the fixed probability vector for an ergodic chain cannot have any 0 entries. | ||
Markov | |||
chain. Let <math>\mat{w}</math> be a fixed probability vector (i.e., <math>\mat{w}</math> is a row | |||
vector | |||
with <math>\mat {w}\mat {P} = \mat {w}</math>). Show that if <math>w_i = 0</math> and <math>p_{ji} > 0</math> | |||
then | |||
<math>w_j = 0</math>. Use this to show that the fixed probability vector for an ergodic | |||
chain | |||
cannot have any 0 entries. | |||
Latest revision as of 22:08, 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]
Let [math]\mat{P}[/math] be the transition matrix of an ergodic Markov chain. Let [math]\mat{w}[/math] be a fixed probability vector (i.e., [math]\mat{w}[/math] is a row vector with [math]\mat {w}\mat {P} = \mat {w}[/math]). Show that if [math]w_i = 0[/math] and [math]p_{ji} \gt 0[/math] then [math]w_j = 0[/math]. Use this to show that the fixed probability vector for an ergodic chain cannot have any 0 entries.