Tight Markov chains and random compositions

TL;DR

This paper proves that certain hitting times in ergodic Markov chains are approximately geometric and independent under specific conditions, enabling precise analysis of large parts in random compositions like column-convex and Carlitz compositions.

Contribution

It establishes a sharp approximation of hitting times by independent geometric and stationary distributions for Markov chains with small transition probabilities, extending to complex compositions.

Findings

Hitting times are nearly geometric and independent for chains with small transition probabilities.

The approximation accurately predicts the distribution of large parts in random compositions.

Limiting distributions of the largest parts are derived for specific composition models.

Abstract

For an ergodic Markov chain $\{X(t)\}$ on $\Bbb N$, with a stationary distribution $\pi$, let $T_n>0$ denote a hitting time for $[n]^c$, and let $X_n=X(T_n)$. Around 2005 Guy Louchard popularized a conjecture that, for $n\to \infty$, $T_n$ is almost Geometric($p$), $p=\pi([n]^c)$, $X_n$ is almost stationarily distributed on $[n]^c$, and that $X_n$ and $T_n$ are almost independent, if $p(n):=\sup_ip(i,[n]^c)\to 0$ exponentially fast. For the chains with $p(n) \to 0$ however slowly, and with $\sup_{i,j}\,\|p(i,\cdot)-p(j,\cdot)\|_{TV}<1$, we show that Louchard's conjecture is indeed true even for the hits of an arbitrary $S_n\subset\Bbb N$ with $\pi(S_n)\to 0$. More precisely, a sequence of $k$ consecutive hit locations paired with the time elapsed since a previous hit (for the first hit, since the starting moment) is approximated, within a total variation distance of order $k\,\sup_ip(i,S_n)$, by a $k$-long sequence of independent copies of $(\ell_n,t_n)$, where $\ell_n= \text{Geometric}\,(\pi(S_n))$, $t_n$ is distributed stationarily on $S_n$, and $\ell_n$ is independent of $t_n$. The two conditions are easily met by the Markov chains that arose in Louchard's studies as likely sharp approximations of two random compositions of a large integer $\nu$, a column-convex animal (cca) composition and a Carlitz (C) composition. We show that this approximation is indeed very sharp for most of the parts of the random compositions. Combining the two approximations in a tandem, we are able to determine the limiting distributions of $\mu=o(\ln\nu)$ and $\mu=o(\nu^{1/2})$ largest parts of the random cca composition and the random C-composition, respectively. (Submitted to Annals of Probability in August, 2009.)