Rate of escape and central limit theorem for the supercritical Lamperti problem

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:120

Article

Menshikov, Mikhail V.²  Wade, Andrew R.¹ 

[1] Univ Strathclyde, Dept Math & Stat, Glasgow G1 1XH, Lanark, Scotland

[2] Univ Durham, Dept Math Sci, Durham DH1 3LE, England

关键词: Lamperti's problem;    Almost-sure bounds;    Law of large numbers;    Central limit theorem;    Birth-and-death chain;    Transience;    Inhomogeneous random walk;   

DOI  :  10.1016/j.spa.2010.06.004

来源: Elsevier

【 摘 要 】

The study of discrete-time stochastic processes on the half-line with mean drift at x given by mu(1)(x) -> 0 as x -> infinity is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where it mu(1)(x) is of order x(-beta) for some beta is an element of (0, 1). The bounds are of order t(1/(1+beta)), so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2 + 2 beta + epsilon)-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where x(beta) mu(1) (x) has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where beta = 0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

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