Coalescence and meeting times on $n$-block Markov chains

TL;DR

This paper studies the asymptotic behavior of coalescence and meeting times in $n$-block Markov chains, revealing a key algebraic quantity that governs their exponential growth rate and its relation to entropy.

Contribution

It introduces an algebraic quantity $L(V,P)$ that characterizes the exponential growth rate of coalescence times in $n$-block Markov chains and links it to the chain's entropy.

Findings

The coalescence time $C_n$ grows exponentially with rate $L(V,P)$.

The quantity $rac{1}{n} ext{log} C_n$ converges to $L(V,P)$ in probability.

A full characterization of the relationship between $L(V,P)$ and the entropy of $(V,P)$.

Abstract

We consider finite state, discrete-time, mixing Markov chains $(V,P)$, where $V$ is the state space and $P$ is transition matrix. To each such chain $(V,P)$, we associate a sequence of chains $(V_n,P_n)$ by coding trajectories of $(V,P)$ according to their overlapping $n$-blocks. The chain $(V_n,P_n)$, called the $n$-block Markov chain associated to $(V,P)$, may be considered an alternate version of $(V,P)$ having memory of length $n$. Along such a sequence of chains, we characterize the asymptotic behavior of coalescence times and meeting times as $n$ tends to infinity. In particular, we define an algebraic quantity $L(V,P)$ depending only on $(V,P)$, and we show that if the coalescence time on $(V_n,P_n)$ is denoted by $C_n$, then the quantity $\frac{1}{n} \log C_n$ converges in probability to $L(V,P)$ with exponential rate. Furthermore, we fully characterize the relationship between $L(V,P)$ and the entropy of $(V,P)$.