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 $\mu ={\mu }_0⊞{\mu }_1⊞\dot{.}.⊞{\mu }_n⊞\dot{.}.$ such that ${\mu }_0$ is infinitely divisible and ${\mu }_i$ is indecomposable for $i\ge 1$. Additionally, we prove that the family of all $⊞$-divisors of a measure $\mu$ is compact up to translation. Analogous results are also proven in the case of multiplicative convolution.

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