exercise:6ee9322c6b: 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 a regular Markov chain. Assume that there are <math>r</math> states and let <math>N(r)</math> be the smallest integer <math>n</math> such that <math>\mat{P}</math> is regular if and only if <m...") |
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\newcommand{\mathds}{\mathbb}</math></div> Let <math>\mat{P}</math> be the transition matrix of a regular | \newcommand{\mathds}{\mathbb}</math></div> Let <math>\mat{P}</math> be the transition matrix of a regular | ||
Markov | Markov chain. Assume that there are <math>r</math> states and let <math>N(r)</math> be the smallest integer <math>n</math> | ||
chain. Assume that there are <math>r</math> states and let <math>N(r)</math> be the smallest | such that <math>\mat{P}</math> is regular if and only if <math>\mat {P}^{N(r)}</math> has no zero entries. Find a finite upper bound for <math>N(r)</math>. See if you can determine <math>N(3)</math> exactly. | ||
integer <math>n</math> | |||
such that <math>\mat{P}</math> is regular if and only if <math>\mat {P}^{N(r)}</math> has no zero | |||
entries. | |||
Find a finite upper bound for <math>N(r)</math>. See if you can determine <math>N(3)</math> exactly. | |||
Revision as of 21:56, 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 a regular
Markov chain. Assume that there are [math]r[/math] states and let [math]N(r)[/math] be the smallest integer [math]n[/math] such that [math]\mat{P}[/math] is regular if and only if [math]\mat {P}^{N(r)}[/math] has no zero entries. Find a finite upper bound for [math]N(r)[/math]. See if you can determine [math]N(3)[/math] exactly.