A Khintchine decomposition for free probability

Abstract

Let $\mu$ be a probability measure on the real line. In this paper we prove that there exists a decomposition $\mu=\mu_{0}\boxplus\mu_{1}\boxplus\cdots\boxplus\mu_{n}\boxplus\cdots$ such that $\mu_{0}$ is infinitely divisible, and $\mu_{i}$ is indecomposable for $i\geq1$. 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.