Excursions and path functionals for stochastic processes with asymptotically zero drifts

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 ${\sum }_{s\le t}{X}_s^{\alpha }$, $\alpha >0$. 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 $\mathbb{R}$ and to a class of multidimensional `centrally biased' random walks on ${\mathbb{R}}^d$; 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.

33 pages

References in corpus (4)

• Localization and delocalization of random interfaces
• Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains
• Random walk weakly attracted to a wall
• Random walk with barycentric self-interaction

Cited by in corpus (8)

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• Extreme value statistics of positive recurrent centrally biased random walks
• At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift
• Non-homogeneous random walks with stochastic resetting: an application to the Gillis model
• Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability
• Superdiffusive planar random walks with polynomial space-time drifts