exercise:4bebd1a462: 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> Define <math>f(r)</math> to be the smallest integer <math>n</math> such that for all regular Markov chains with <math>r</math> states, the <math>n</math>th power of the transition matrix has all entries positive. It has been shown,<ref group="Not...")
 
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\newcommand{\mathds}{\mathbb}</math></div> Define <math>f(r)</math> to be the smallest integer <math>n</math> such
\newcommand{\mathds}{\mathbb}</math></div> Define <math>f(r)</math> to be the smallest integer <math>n</math> such that for all regular Markov chains with <math>r</math> states, the <math>n</math>th power of the transition matrix has all entries positive.  It has been shown,<ref group="Notes" >E. Seneta, ''Non-Negative Matrices:  An Introduction to Theory and Applications,''
that
Wiley, New York, 1973, pp. 52-54.</ref> that <math>f(r) = r^2 - 2r + 2</math>.   
for all regular Markov chains with <math>r</math> states, the <math>n</math>th power of the
<ul style="list-style-type:lower-alpha"><li>
transition
matrix has all entries positive.  It has been shown,<ref group="Notes" >E. Seneta, ''Non-Negative Matrices:  An Introduction to Theory and Applications,''
Wiley, New York, 1973, pp. 52-54.</ref>
that <math>f(r) = r^2 - 2r + 2</math>.   
<ul><li>
Define the transition matrix of an <math>r</math>-state Markov chain as follows:
Define the transition matrix of an <math>r</math>-state Markov chain as follows:
For states <math>s_i</math>, with <math>i = 1</math>, 2, \ldots, <math>r - 2</math>,  
For states <math>s_i</math>, with <math>i = 1</math>, 2, \ldots, <math>r - 2</math>,  

Latest revision as of 22:58, 17 June 2024

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Define [math]f(r)[/math] to be the smallest integer [math]n[/math] such that for all regular Markov chains with [math]r[/math] states, the [math]n[/math]th power of the transition matrix has all entries positive. It has been shown,[Notes 1] that [math]f(r) = r^2 - 2r + 2[/math].

  • Define the transition matrix of an [math]r[/math]-state Markov chain as follows: For states [math]s_i[/math], with [math]i = 1[/math], 2, \ldots, [math]r - 2[/math], [math]\mat {P}(i,i + 1) = 1[/math], [math]\mat {P}(r - 1,r) = \mat {P}(r - 1, 1) = 1/2[/math], and [math]\mat {P}(r,1) = 1[/math]. Show that this is a regular Markov chain.
  • For [math]r = 3[/math], verify that the fifth power is the first power that has no zeros.
  • Show that, for general [math]r[/math], the smallest [math]n[/math] such that [math]\mat {P}^n[/math] has all entries positive is [math]n = f(r)[/math].

Notes

  1. E. Seneta, Non-Negative Matrices: An Introduction to Theory and Applications, Wiley, New York, 1973, pp. 52-54.