The variation threshold, Kemeny's constant and stationary times of reversible Markov chains
Year: 2026
Type: preprint
Abstract: For finite reversible Markov chains, Aldous and Fill asked whether the variation threshold $\tau_1$ is at most the average hitting time $\tau_0$ (Kemeny's constant), and whether the relaxation time $\tau_2$ is at most $\tau_1^{(2)}$, the largest over initial states of the least mean of a stopping time at which the chain has its stationary distribution. The answer to the first question is no: a birth and death chain on three states with a strong drift has $\tau_1>\tau_0$, and on the analogous chains with $k+1$ states the ratio $\tau_1/\tau_0$ tends to $t_k/k$ as the drift grows, where $t_k$ is the point at which the survival function of the Gamma$(k,1)$ distribution equals $e^{-1}$; for $k=4$ this limit is $1.0879\ldots$. The answer to the second question is yes, in continuous and in discrete time.
Author: Shmalo, Yitzchak
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225653687
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