TIGHT MARKOV CHAINS AND RANDOM COMPOSITIONS

• 成果类型: Article
• 署名作者: Pittel, Boris
• 署名单位: University System of Ohio; Ohio State University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP656
• 发表日期: 2012
• 页码: 1535-1576
• 关键词: diagonally-convex animals; probabilistic analysis; carlitz compositions; column-convex; integer

摘要

For an ergodic Markov chain {X (t)} on 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 -> infinity, 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(i) p(i,[n](c)) -> 0 exponentially fast. For the chains with p(n) -> 0, however slowly, and with sup(i,j) parallel to p(i, .) - p(j, .)parallel to Tv < 1, we show that Louchard's conjecture is indeed true, even for the hits of an arbitrary S-n subset of N with pi (S-n) -> 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(i) p(i, S-n), by a k-long sequence of independent copies of (l(n), t(n)), where t(n) = Geometric(pi(S-n)), l(n) is distributed stationarily on S-n and E-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 In-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 (l(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.)