Some limit theorems for heights of random walks on spider

TL;DR

This paper studies the behavior of simple symmetric random walks on a spider graph, establishing limit theorems for the heights reached on the legs, and analyzing how these heights behave as the number of legs increases.

Contribution

It introduces limit theorems for the heights of random walks on spider graphs and explores their behavior with varying numbers of legs, including strong approximations by Brownian spiders.

Findings

Strong approximation of the random walk by Brownian spider.

Limit theorems for maximum heights on the legs.

Analysis of height behavior as the number of legs grows.

Abstract

A simple symmetric random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker can go on the legs in $n$ steps. The heights on the legs are also investigated when the number of legs goes to infinity.