A note on concentration functions on transformation groups

TL;DR

This paper investigates the behavior of concentration functions for group actions, establishing a precise criterion for their convergence to zero in the context of spread-out irreducible measures on locally compact groups.

Contribution

It provides a necessary and sufficient condition for the convergence of concentration functions in the setting of continuous group actions on locally compact spaces.

Findings

Established a criterion for concentration function convergence

Connected measure properties with topological group actions

Extended understanding of probabilistic behavior in group dynamics

Abstract

In this note, we consider the concentration function problem for a continuous action of a locally compact group $G$ on a locally compact Hausdorff space $X$. We prove a necessary and sufficient condition for the concentration functions of a spread-out irreducible probability measure $\mu$ on $G$ to converge to zero.