A combinatorial perspective on the Kemeny constant and more
Luis Fredes, Jean-François Marckert
TL;DR
The paper generalizes Kemeny’s constant through a new functional identity that ties hitting-time generating functions to determinants of principal minors of the transition matrix. By introducing ${\bf K}_u^{\geq t}(x)=\sum_v G_{u,v}^{\geq t}(x)\pi_v(x)$ and proving that these power series are independent of the starting state $u$ (with ${\bf K}^{\geq t}(x)=x^t{\bf K}^{\geq 0}(x)$ and ${\bf K}^{\geq 0}(x)=\det({\sf Id}_d-xM)/(1-x)$), the authors obtain a robust, algebraic framework for analyzing first-visit times. This approach yields further results on recursion relations for hitting-time laws, factorial-moment identities, alternative matrix representations, and eigenvalue-based characterizations of the Kemeny constant, providing new insights beyond the classical constant. The combination of probabilistic stopping-time arguments with exact linear-algebraic identities offers a powerful tool for understanding the structure of hitting times in finite Markov chains and connects to established concepts such as the invariant measure, adjugate relations, and the group inverse. The paper also demonstrates the theory through an explicit example, illustrating the interplay between generating functions, determinants, and hitting-time statistics with potential applications to stochastic processes and network analysis.
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 $τ^{\geq 1}_u$ denote the first positive hitting time of $u$ by $C$, and $ρ$ the unique invariant measure of $M$. Kemeny proved that if $X$ is sampled according to $ρ$ 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 $(E(τ^{\geq 1}_X~|~C_0=u), u \in V)$ are equal. \par In this paper, we show that this property follows from a more general result: the generating function $\sum_{v\in V}E(x^{τ_v^{\geq 1}}~|~C_0=u)\det(Id-xM^{(v)})$ 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 $τ^{\geq 1}_v$, 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 $(ρ_u,u\in V)$. From this property, we deduce several further results, including relations involving higher moments of $τ_X^{\geq 1}$, which are of independent interest.