A Khintchine Decomposition for Free Probability

TL;DR

This paper establishes a decomposition theorem for probability measures in free probability, showing measures can be broken into infinitely divisible and indecomposable parts, with results extended to multiplicative convolution.

Contribution

It introduces a Khintchine-type decomposition for free probability measures and proves the compactness of their divisors, extending classical results to free and multiplicative convolutions.

Findings

Existence of a free probability measure decomposition into infinitely divisible and indecomposable parts.

Compactness of the set of all divisors of a measure up to translation.

Extension of decomposition results to multiplicative free convolution.

Abstract

Let $\mu$ be a probability measure on the real line. In this paper we prove that there exists a decomposition $\mu = \mu_{0} \boxplus \mu_{1} \boxplus \... \boxplus \mu_{n} \boxplus \...$ such that $\mu_{0}$ is infinitely divisible and $\mu_{i}$ is indecomposable for $i \geq 1$. Additionally, we prove that the family of all $\boxplus$-divisors of a measure $\mu$ is compact up to translation. Analogous results are also proven in the case of multiplicative convolution.