The Annals of Probability

Tight Markov chains and random compositions

Boris Pittel

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Abstract

For an ergodic Markov chain $\{X(t)\}$ on $\mathbb{N}$, 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 $\operatorname{Geometric}(p)$, $p=\pi([n]^{c})$, $X_{n}$ is almost stationarily distributed on $[n]^{c}$ and that $X_{n}$ and $T_{n}$ are almost independent, if $p(n):=\sup_{i}p(i,[n]^{c})\to0$ exponentially fast. For the chains with $p(n)\to0$, however slowly, and with $\sup_{i,j}\|p(i,\cdot)-p(j,\cdot)\|_{\mathrm{TV}}<1$, we show that Louchard’s conjecture is indeed true, even for the hits of an arbitrary $S_{n}\subset\mathbb{N}$ with $\pi(S_{n})\to0$. 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 $k\sup_{i}p(i,S_{n})$, by a $k$-long sequence of independent copies of $(\ell_{n},t_{n})$, where $t_{n}=\operatorname{Geometric}(\pi(S_{n}))$, $\ell_{n}$ 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 each of the random compositions, read from left to right, for as long as the sum of the remaining parts stays above $\ln^{2}\nu$. Combining the two approximations, a composition—by its chain, and, for $S_{n}=[n]^{c}$, the sequence of hit locations paired each with a time elapsed from the previous hit—by the independent copies of $(\ell_{n},t_{n})$, enables us 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 June 2009.)

Article information

Source
Ann. Probab., Volume 40, Number 4 (2012), 1535-1576.

Dates
First available in Project Euclid: 4 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1341401143

Digital Object Identifier
doi:10.1214/11-AOP656

Mathematical Reviews number (MathSciNet)
MR2978132

Zentralblatt MATH identifier
1257.05009

Subjects
Primary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A17: Partitions of integers [See also 11P81, 11P82, 11P83] 11P99: None of the above, but in this section 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 60J05: Discrete-time Markov processes on general state spaces

Keywords
Random compositions Markov chains rarity exponentiality extreme values limit theorems

Citation

Pittel, Boris. Tight Markov chains and random compositions. Ann. Probab. 40 (2012), no. 4, 1535--1576. doi:10.1214/11-AOP656. https://projecteuclid.org/euclid.aop/1341401143


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