Excursions and path functionals for stochastic processes with asymptotically zero drifts

TL;DR

This paper analyzes stochastic processes with asymptotically zero drifts, providing sharp asymptotics for functionals related to excursions, maxima, and return times, with applications to polymers, random walks, and urn models.

Contribution

It offers new asymptotic results for functionals of processes with zero mean drift, improving existing literature and extending to multidimensional and specific models.

Findings

Sharp asymptotics for excursion-related functionals.

New results on sums of process powers and maxima.

Applications to random polymers, urns, and biased random walks.

Abstract

We study discrete-time stochastic processes $(X_t)$ on $[0,\infty)$ with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at $x$ is about $c/x$. Our focus is the recurrent case (when $c$ is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form $\sum_{s \leq t} X_s^\alpha$, $\alpha >0$. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of $\R$ and to a class of multidimensional `centrally biased' random walks on $\R^d$; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al.