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. 20782099. ISSN 03044149 (https://doi.org/10.1016/j.spa.2010.06.004)
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Abstract
The study of discretetime stochastic processes on the halfline with mean drift at x given by μ1(x)→0 as x→∞ is known as Lamperti's problem. We give sharp almostsure 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 superdiffusive but subballistic (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 timehomogeneous 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 nearestneighbour 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 birthanddeath chains and to multidimensional nonhomogeneous random walks.


Item type: Article ID code: 27224 Dates: DateEventSeptember 2010PublishedKeywords: lamperti’s problem, almostsure bounds, law of large numbers, central limit theorem, birthanddeath chain, transience, inhomogeneous random walk, Mathematics, Modelling and Simulation, Applied Mathematics, Statistics and Probability Subjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Mrs Carolynne Westwood Date deposited: 30 Aug 2010 14:27 Last modified: 25 Aug 2021 01:10 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/27224