The hitting distribution of a line segment for two dimensional random walks

TL;DR

This paper provides asymptotic estimates for the probability distribution of a two-dimensional random walk hitting a long segment on the real axis, revealing similarities to Brownian motion and detailed walk characteristics.

Contribution

It offers new asymptotic estimates for hitting distributions of 2D random walks, connecting them to Brownian density and incorporating walk-specific features.

Findings

Asymptotic estimates resemble Brownian density

Results include detailed walk characteristics

Some estimates are general, others detailed

Abstract

Asymptotic estimates of the hitting distribution of a long segment on the real axis for two dimensional random walks on ${\bf Z}^2$ of zero mean and finite variances are obtained: some are general and exhibit its apparent similarity to the corresponding Brownian density, while others are so detailed as to involve certain characteristics of the random walk.