Random walks driven by low moment measures

TL;DR

This paper investigates the decay rates of convolution powers of probability measures on groups that lack a second moment but satisfy weaker moment conditions, introducing a function to describe the fastest decay possible.

Contribution

It introduces a new function $\Phi_{G, ho}$ that characterizes decay rates for measures with weaker moments on unimodular groups, providing estimates for various groups and functions.

Findings

Estimated decay functions for different groups and measures

Established $\Phi_{G, ho}$ as a group invariant under certain conditions

Analyzed decay behavior without requiring second moments

Abstract

We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group G and any positive function $\rho:G \rightarrow [0,+\infty]$, we introduce a function $\Phi_{G,\rho}$ which describes the fastest possible decay of $n \mapsto \phi^{(2n)}(e)$ when \phi is a symmetric continuous probability density such that $\int\rho\phi$ is finite. We estimate $\Phi_{G,\rho}$ for a variety of groups G and functions \rho. When \rho is of the form $\rho=\rho \circ \delta$ with $\rho:[0,+\infty) \rightarrow [0,+\infty)$, a fixed increasing function, and $\delta:G \rightarrow [0,+\infty)$, a natural word length measuring the distance to the identity element in G, $\Phi_{G,\rho}$ can be thought of as a group invariant.