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July 2012 Tight Markov chains and random compositions
Boris Pittel
Ann. Probab. 40(4): 1535-1576 (July 2012). DOI: 10.1214/11-AOP656

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.)

Citation

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Boris Pittel. "Tight Markov chains and random compositions." Ann. Probab. 40 (4) 1535 - 1576, July 2012. https://doi.org/10.1214/11-AOP656

Information

Published: July 2012
First available in Project Euclid: 4 July 2012

zbMATH: 1257.05009
MathSciNet: MR2978132
Digital Object Identifier: 10.1214/11-AOP656

Subjects:
Primary: 05A15 , 05A17 , 11P99 , 60C05 , 60F05 , 60J05

Keywords: exponentiality , Extreme values , limit theorems , Markov chains , Random compositions , rarity

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 4 • July 2012
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