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

Menshikov, Mikhail V. and Wade, A.R. (2010) Rate of escape and central limit theorem for the supercritical Lamperti problem. Stochastic Processes and their Applications, 120 (10). pp. 2078-2099. ISSN 0304-4149

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    Abstract

    The study of discrete-time stochastic processes on the half-line with mean drift at x given by μ1(x)→0 as x→∞ is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where μ1(x) is of order x−β for some β(0,1). The bounds are of order t1/(1+β), 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+2β+ε)-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 xβμ1(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 β=0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.

    Item type: Article
    ID code: 27224
    Keywords: lamperti’s problem, almost-sure bounds, law of large numbers, central limit theorem, birth-and-death chain, transience, inhomogeneous random walk, Mathematics, Modelling and Simulation, Applied Mathematics, Statistics and Probability
    Subjects: Science > Mathematics
    Department: Faculty of Science > Mathematics and Statistics
    Related URLs:
    Depositing user: Mrs Carolynne Westwood
    Date Deposited: 30 Aug 2010 15:27
    Last modified: 27 Mar 2014 11:24
    URI: http://strathprints.strath.ac.uk/id/eprint/27224

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