Transient NN random walk on the line

TL;DR

This paper establishes strong theorems about the local time at infinity for a transient nearest neighbor random walk on the line, including laws of the iterated logarithm and interval length analysis.

Contribution

It provides new rigorous results on the asymptotic behavior of local time and interval traversal lengths for transient one-dimensional random walks.

Findings

Laws of the iterated logarithm for large local times

Characterization of interval lengths without return

Insights into the walk's asymptotic properties

Abstract

We prove strong theorems for the local time at infinity of a nearest neighbor transient random walk. First, laws of the iterated logarithm are given for the large values of the local time. Then we investigate the length of intervals over which the walk runs through (always from left to right) without ever returning.