Taylor expansions of R-transforms, application to supports and moments

Abstract

We prove that a probability measure on the real line has a moment of order p (even integer), if and only if its R-transform admits a Taylor expansion with p terms. We also prove a weaker version of this result when p is odd. Then, we apply this to prove that a probability measure whose R-transform extends analytically to a ball with center zero is compactly supported, and that a free infinitely divisible distribution has a moment of order p even, if and only if its Levy measure does so. We also prove a weaker version of the last result when p is odd.