A Law of Iterated Logarithm on Lamplighter Diagonal Products

TL;DR

This paper establishes a Law of Iterated Logarithm for random walks on certain diagonal product groups, revealing new behaviors and examples of growth rates for these stochastic processes.

Contribution

It proves a Law of Iterated Logarithm for random walks on diagonal product groups, expanding the understanding of their asymptotic behaviors.

Findings

Existence of groups with random walks exhibiting specific growth rates

Demonstration of LIL behaviors for a range of exponents 1/2 to 1

Construction of examples with precise limsup and liminf properties

Abstract

We prove a Law of Iterated Logarithm for random walks on a family of diagonal products constructed by Brieussel and Zheng (2021). This provides a wide variety of new examples of Law of Iterated Logarithm behaviours for random walks on groups. In particular, it follows that for any $\frac{1}{2}\leq \beta\leq 1$ there is a group $G$ and random walk $W_n$ on $G$ with $\mathbb{E}|W_n|\simeq n^\beta$ such that $$0<\limsup \frac{|W_n|}{n^\beta(\log\log n)^{1-\beta}}<\infty$$ and $$0<\liminf \frac{|W_n|(\log\log n)^{1-\beta}}{n^\beta}<\infty.$$