What is typical?

TL;DR

This paper extends the concept of typicality from compact to locally compact groups, linking it with mass-stationarity and advancing Palm theory by providing a natural definition and demonstrating its invariance under random shifts.

Contribution

It introduces a natural definition of typicality for random measures on locally compact groups and connects it with mass-stationarity, expanding the theoretical framework of Palm theory.

Findings

Defined typicality in the compact case and extended it to locally compact groups.

Proved invariance of typicality under random shifts around the origin.

Connected the concept with mass-stationarity in the broader setting.

Abstract

Let $\xi$ be a random measure on a locally compact second countable topological group and let $X$ be a random element in a measurable space on which the group acts. In the compact case, we give a natural definition of the concept that the origin is a typical location for $X$ in the mass of $\xi$, and prove that when this holds the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.