Kellogg's theorem

In probability theory, Kemeny’s constant is the expected number of time steps required for a Markov chain to transition from a starting state i to a random destination state sampled from the Markov chain's stationary distribution. Surprisingly, this quantity does not depend on which starting state i is chosen. It is in that sense a constant, although it is different for different Markov chains. When first published by John Kemeny in 1960 a prize was offered for an intuitive explanation as to why quantity was constant.

Definition

For a finite ergodic Markov chain with transition matrix P and invariant distribution π, write m_ij for the mean first passage time from state i to state j (denoting the mean recurrence time for the case i = j). Then

$K=\sum _j{\pi }_j{m}_{ij}$

is a constant and not dependent on i.

Prize

Kemeny wrote, (for i the starting state of the Markov chain) “A prize is offered for the first person to give an intuitively plausible reason for the above sum to be independent of i.” Grinstead and Snell offer an explanation by Peter Doyle as an exercise, with solution “he got it!”

In the course of a walk with Snell along Minnehaha Avenue in Minneapolis in the fall of 1983, Peter Doyle suggested the following explanation for the constancy of Kemeny's constant. Choose a target state according to the fixed vector w. Start from state i and wait until the time T that the target state occurs for the first time. Let K_i be the expected value of T. Observe that

$K_i+w_i\cdot 1/w_i=\sum _j{P}_{ij}{K}_j+1$

and hence

$K_i=\sum _j{P}_{ij}{K}_j.$

By the maximum principle, K_i is a constant. Should Peter have been given the prize?

References

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