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

TL;DR

This paper investigates the asymptotic behavior of supercritical Lamperti processes, establishing sharp bounds, a law of large numbers, and a central limit theorem under minimal assumptions, with applications to various stochastic models.

Contribution

The paper provides the first sharp almost-sure bounds, a strong law of large numbers, and a new central limit theorem for Lamperti processes with minimal moment conditions, extending previous results.

Findings

Sharp bounds of order t^{1/(1+β)} for the process

A strong law of large numbers under finite positive limit conditions

A new central limit theorem applicable to non-Markovian processes

Abstract

The study of discrete-time stochastic processes on the half-line with mean drift at $x$ given by $\mu_1 (x) \to 0$ as $x \to \infty$ is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where $\mu_1 (x)$ is of order $x^{-\beta}$ for some $\beta \in (0,1)$. 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 $(2+2\beta+\varepsilon)$-moments for our main results, so 4th 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.