Centering problems for probability measures on finite dimensional vector spaces

TL;DR

This paper investigates centering problems for probability measures on finite-dimensional vector spaces, establishing existence and explicit forms of centering vectors, especially for infinitely divisible and quasi-decomposable measures, extending prior results.

Contribution

It generalizes Jurek's result by proving the existence of a universal centering vector for all measures and provides explicit forms for infinitely divisible measures, analyzing conditions for quasi-decomposable measures.

Findings

Existence of a centering vector for all probability measures on finite-dimensional spaces.

Explicit form of the centering vector for infinitely divisible measures.

Conditions for universal centering of quasi-decomposable measures.

Abstract

The paper deals with various centering problems for probability measures on finite dimensional vector spaces. We show that for every such measure there exists a vector $h$ satisfying $\mu*\delta(h)=S(\mu*\delta (h))$ for each symmetry $S$ of $\mu$, generalizing thus Jurek's result obtained for full measures. An explicit form of the $h$ is given for infinitely divisible $\mu$. The main result of the paper consists in the analysis of quasi-decomposable (operator-semistable and operator-stable) measures and finding conditions for the existence of a `universal centering' of such a measure to a strictly quasi-decomposable one.