exercise:08e8abf24d: 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> In the course of a walk with Snell along Minnehaha Avenue in Minneapolis in the fall of 1983, Peter Doyle<ref group="Notes" >Private communication.</ref> suggested the following explanation for the constancy of ''Kemeny's constant'' (see Exercise...") |
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\newcommand{\mathds}{\mathbb}</math></div> In the course of a walk with Snell along Minnehaha Avenue in Minneapolis | \newcommand{\mathds}{\mathbb}</math></div> In the course of a walk with Snell along Minnehaha Avenue in Minneapolis in the fall of 1983, Peter Doyle<ref group="Notes" >Private communication.</ref> suggested the following explanation for the constancy of ''Kemeny's constant'' (see [[exercise:F040c28e4e |Exercise]]). Choose a target state according to the fixed vector <math>\mat{w}</math>. Start from state <math>i</math> and wait until the time <math>T</math> that the target state occurs for the first time. Let <math>K_i</math> be the expected value | ||
in the fall of 1983, Peter Doyle<ref group="Notes" >Private | |||
communication.</ref> suggested the following explanation for the constancy of | |||
''Kemeny's | |||
constant'' (see | |||
to the fixed vector <math>\mat{w}</math>. Start from state <math>i</math> and wait until the time <math>T</math> that | |||
the target state occurs for the first time. Let <math>K_i</math> be the expected value | |||
of <math>T</math>. Observe that | of <math>T</math>. Observe that | ||
Latest revision as of 02:31, 15 June 2024
[math]
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In the course of a walk with Snell along Minnehaha Avenue in Minneapolis in the fall of 1983, Peter Doyle[Notes 1] suggested the following explanation for the constancy of Kemeny's constant (see Exercise). Choose a target state according to the fixed vector [math]\mat{w}[/math]. Start from state [math]i[/math] and wait until the time [math]T[/math] that the target state occurs for the first time. Let [math]K_i[/math] be the expected value
of [math]T[/math]. Observe that
[[math]]
K_i + w_i \cdot 1/w_i= \sum_j P_{ij} K_j + 1\ ,
[[/math]]
and hence
[[math]]
K_i = \sum_j P_{ij} K_j\ .
[[/math]]
By the maximum principle, [math]K_i[/math] is a constant. Should Peter have been given the prize?
Notes