Tight Markov chains and random compositions

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

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.)

First Page:
Primary Subjects: 05A15, 05A17, 11P99, 60C05, 60F05, 60J05
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1341401143
Digital Object Identifier: doi:10.1214/11-AOP656
Zentralblatt MATH identifier: 06067450
Mathematical Reviews number (MathSciNet): MR2978132

References

[1] Aldous, D. J. (1982). Markov chains with almost exponential hitting times. Stochastic Process. Appl. 13 305–310.
Mathematical Reviews (MathSciNet): MR671039
Digital Object Identifier: doi:10.1016/0304-4149(82)90016-3
[2] Aldous, D. J. and Brown, M. (1992). Inequalities for rare events in time-reversible Markov chains. I. In Stochastic Inequalities (Seattle, WA, 1991). Institute of Mathematical Statistics Lecture Notes—Monograph Series 22 1–16. IMS, Hayward, CA.
Mathematical Reviews (MathSciNet): MR1228050
Digital Object Identifier: doi:10.1214/lnms/1215461937
[3] Aldous, D. J. and Brown, M. (1993). Inequalities for rare events in time-reversible Markov chains. II. Stochastic Process. Appl. 44 15–25.
Mathematical Reviews (MathSciNet): MR1198660
Digital Object Identifier: doi:10.1016/0304-4149(93)90035-3
[4] Andrews, G. E. (1976). The Theory of Partitions. Encyclopedia of Mathematics and Its Applications 2. Addison-Wesley, Reading, MA.
Mathematical Reviews (MathSciNet): MR557013
[5] Bender, E. A. (1973). Central and local limit theorems applied to asymptotic enumeration. J. Combinatorial Theory Ser. A 15 91–111.
Mathematical Reviews (MathSciNet): MR375433
Digital Object Identifier: doi:10.1016/0097-3165(73)90038-1
[6] Bender, E. A. and Canfield, E. R. (2005). Locally restricted compositions. I. Restricted adjacent differences. Electron. J. Combin. 12 1–27 (electronic).
Mathematical Reviews (MathSciNet): MR2180794
Mathematical Reviews (MathSciNet): MR229267
[8] Carlitz, L. (1976). Restricted compositions. Fibonacci Quart. 14 254–264.
Mathematical Reviews (MathSciNet): MR414479
[9] Derman, C. (1954). A solution to a set of fundamental equations in Markov chains. Proc. Amer. Math. Soc. 5 332–334.
Mathematical Reviews (MathSciNet): MR60757
Digital Object Identifier: doi:10.1090/S0002-9939-1954-0060757-0
[10] Durrett, R. (2005). Probability: Theory and Examples, 3rd ed. Brooks/Cole, Thomson Learning, Belmont, CA.
Mathematical Reviews (MathSciNet): MR1068527
[11] Hitczenko, P. and Louchard, G. (2001). Distinctness of compositions of an integer: A probabilistic analysis. Random Structures Algorithms 19 407–437.
Mathematical Reviews (MathSciNet): MR1871561
[12] Hitczenko, P. and Savage, C. D. (2004). On the multiplicity of parts in a random composition of a large integer. SIAM J. Discrete Math. 18 418–435 (electronic).
Mathematical Reviews (MathSciNet): MR2112515
Digital Object Identifier: doi:10.1137/S0895480199363155
[13] Kac, M. (1947). On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53 1002–1010.
Mathematical Reviews (MathSciNet): MR22323
Digital Object Identifier: doi:10.1090/S0002-9904-1947-08927-8
Project Euclid: euclid.bams/1183511152
[14] Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality. Applied Mathematical Sciences 28. Springer, New York.
Mathematical Reviews (MathSciNet): MR528293
[15] Klarner, D. A. (1965). Some results concerning polyominoes. Fibonacci Quart. 3 9–20.
Mathematical Reviews (MathSciNet): MR186569
[16] Knopfmacher, A. and Prodinger, H. (1998). On Carlitz compositions. European J. Combin. 19 579–589.
Mathematical Reviews (MathSciNet): MR1637748
Digital Object Identifier: doi:10.1006/eujc.1998.0216
[17] Knuth, D. E., Motwani, R. and Pittel, B. (1990). Stable husbands. Random Structures Algorithms 1 1–14.
Mathematical Reviews (MathSciNet): MR1068488
[18] Louchard, G. (2005). Private communication.
[19] Louchard, G. (1997). Probabilistic analysis of column-convex and directed diagonally-convex animals. Random Structures Algorithms 11 151–178.
Mathematical Reviews (MathSciNet): MR1610265
[20] Louchard, G. (1999). Probabilistic analysis of column-convex and directed diagonally-convex animals. II. Trajectories and shapes. Random Structures Algorithms 15 1–23.
Mathematical Reviews (MathSciNet): MR1698406
[21] Louchard, G. and Prodinger, H. (2002). Probabilistic analysis of Carlitz compositions. Discrete Math. Theor. Comput. Sci. 5 71–95 (electronic).
Mathematical Reviews (MathSciNet): MR1902415
[22] Privman, V. and Forgács, G. (1987). Exact solution of the partially directed compact lattice animal model. J. Phys. A 20 L543–L547.
Mathematical Reviews (MathSciNet): MR893296
Digital Object Identifier: doi:10.1088/0305-4470/20/8/011
[23] Privman, V. and Švrakić, N. M. (1988). Exact generating function for fully directed compact lattice animals. Phys. Rev. Lett. 60 1107–1109.
Mathematical Reviews (MathSciNet): MR932171
Digital Object Identifier: doi:10.1103/PhysRevLett.60.1107