Construction and Characterisation of Stationary and Mass-Stationary Random Measures on ${\mathbb R}^d$

TL;DR

This paper develops methods to construct and characterize stationary and mass-stationary random measures on Euclidean space, providing new insights into their intrinsic properties and invariance under shifts.

Contribution

It introduces novel constructions of stationary and mass-stationary measures via change of measure and origin, and characterizes mass-stationarity through distributional invariance under preserving shifts.

Findings

Constructed stationary and mass-stationary measures using change of measure.

Provided characterizations of mass-stationarity via invariance under shifts.

Clarified the relationship between stationarity and mass-stationarity.

Abstract

Mass-stationarity means that the origin is at a typical location in the mass of a random measure. It is an intrinsic characterisation of Palm versions with respect to stationary random measures. Stationarity is the special case when the random measure is Lebesgue measure. The paper presents constructions of stationary and mass-stationary versions through change of measure and change of origin. Further, the paper considers characterisations of mass-stationarity by distributional invariance under preserving shifts agains stationary independent backgrounds.