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 21:58, 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