A combinatorial perspective on the Kemeny constant and more

TL;DR

This paper provides a combinatorial perspective on the Kemeny constant, showing that certain generating functions related to hitting times in Markov chains are independent of initial states, leading to new insights and relations involving moments.

Contribution

It generalizes Kemeny's result by demonstrating the independence of a generating function involving hitting times and determinants from the initial state in Markov chains.

Findings

The generating function sum is independent of the starting state.

Derived relations involving higher moments of hitting times.

Connected determinants to the invariant measure.

Abstract

Let $M$ be an irreducible transition matrix on a finite state space $V$. For a Markov chain $C=(C_k,k\geq 0)$ with transition matrix $M$, let $\tau^{\geq 1}_u$ denote the first positive hitting time of $u$ by $C$, and $\rho$ the unique invariant measure of $M$. Kemeny proved that if $X$ is sampled according to $\rho$ independently of $C$, the expected value of the first positive hitting time of $X$ by $C$ does not depend on the starting state of the chain: all the values $(E(\tau^{\geq 1}_X~|~C_0=u), u \in V)$ are equal. \par In this paper, we show that this property follows from a more general result: the generating function $\sum_{v\in V}E(x^{\tau_v^{\geq 1}}~|~C_0=u)\det(Id-xM^{(v)})$ is independent of the starting state $u$, where $M^{(v)}$ is obtained from $M$ by deleting the row and column corresponding to the state $v$. The factors appearing in this generating function are: first, the probability generating function of $\tau^{\geq 1}_v$, and second, the sequence of determinants $(det(Id-xM^{(v)}),v\in V),$ which, for $x=1$, is known to be proportional to the invariant measure $(\rho_u,u\in V)$. From this property, we deduce several further results, including relations involving higher moments of $\tau_X^{\geq 1}$, which are of independent interest.