Transient nearest neighbor random walk and Bessel process

TL;DR

This paper establishes a strong invariance principle linking a transient Bessel process with a constructed nearest neighbor random walk, demonstrating their local times' similarity and implications for strong limit theorems.

Contribution

It introduces a novel strong invariance principle between Bessel processes and NN random walks, and explores their local times and distributional closeness.

Findings

Strong invariance principle between Bessel process and NN random walk

Local times of both processes are closely aligned

Results on the closeness of NN random walks with similar distributions

Abstract

We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. It is also shown that their local times are close enough to share the same strong limit theorems. It is shown furthermore, that if the difference between the distributions of two NN random walks are small, then the walks themselves can be constructed so that they are close enough. Finally, some consequences concerning strong limit theorems are discussed.