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

Authors: Florent Benaych-Georges

Issue: Volume 55 (2006), Issue 2, 465-482

Abstract: We prove that a probability measure on the real line has a moment of order (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. We then 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 even order $p$, if and only if its L\'evy measure does so. We also prove a weaker version of the last result when $p$ is odd.