Strong limit theorems for a simple random walk on the 2-dimensional comb

TL;DR

This paper investigates the behavior of a simple random walk on a 2D comb lattice, establishing strong limit theorems and laws of the iterated logarithm for its components, revealing its asymptotic properties.

Contribution

The paper introduces a strong approximation result for the random walk on the 2D comb, enabling new limit theorems and asymptotic analyses specific to this lattice structure.

Findings

Established a strong approximation for the walk on the comb

Proved joint Strassen type law of the iterated logarithm

Demonstrated marginal Hirsch type behavior

Abstract

We study the path behaviour of a simple random walk on the 2-dimensional comb lattice ${\mathbb C}^2$ that is obtained from ${\mathbb Z}^2$ by removing all horizontal edges off the x-axis. In particular, we prove a strong approximation result for such a random walk which, in turn, enables us to establish strong limit theorems, like the joint Strassen type law of the iterated logarithm of its two components, as well as their marginal Hirsch type behaviour.