Asymptotics of returns to the coordinate hyperplanes for conditioned simple random walks

TL;DR

This paper investigates the asymptotic behavior of returns to coordinate hyperplanes in multidimensional random walks under various conditionings, revealing diverse limiting distributions and independence properties.

Contribution

It provides the first detailed asymptotic analysis of hyperplane returns in higher dimensions, characterizing limiting distributions and independence structures.

Findings

Limiting distributions include half-normal, Rayleigh, geometric, and negative binomial.

Coordinates are asymptotically independent in most cases.

Notable exceptions occur in the meander case depending on drift.

Abstract

In this paper we study the number of returns to the coordinate hyperplanes for multidimensional nearest-neighbour random walks. While one-dimensional results on returns are classical, much less is known in higher dimensions. We analyse the asymptotic behaviour of returns under several natural conditionings: the unconditioned walk, bridges, meanders, and non-negative bridges (or excursions). Our main results characterize the limiting distributions under appropriate rescaling. The resulting one-dimensional marginals may be half-normal, Rayleigh, geometric, negative binomial, or certain mixtures thereof. In most situations, the coordinates are asymptotically independent; however, there are notable exceptions for the meander case, depending on the drift. The proofs rely on conditioning on the numbers of horizontal and vertical steps, which restores a form of independence and reduces the problem to one-dimensional estimates via binomial convolution and Bernstein-type approximations.