Skip to main content

Research Repository

Advanced Search

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

Menshikov, M.V.; Wade, Andrew R.

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



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.


Menshikov, M., & Wade, A. R. (2010). Rate of escape and central limit theorem for the supercritical Lamperti problem. Stochastic Processes and their Applications, 120(10), 2078-2099.

Journal Article Type Article
Publication Date Sep 1, 2010
Deposit Date Mar 1, 2011
Publicly Available Date Jan 31, 2013
Journal Stochastic Processes and their Applications
Print ISSN 0304-4149
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 120
Issue 10
Pages 2078-2099
Keywords Lamperti’s problem, Almost-sure bounds, Law of large numbers, Central limit theorem, Birth-and-death chain, Transience, Inhomogeneous random walk.


Accepted Journal Article (358 Kb)

Copyright Statement
NOTICE: this is the author’s version of a work that was accepted for publication in Stochastic processes and their applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Stochastic processes and their applications, 120(10), 2010, 10.1016/

You might also like

Downloadable Citations