Convolution and Limit Theorems for Conditionally Free Random Variables

Abstract

We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its $R$-transform. We calculate explicitly the distributions of the conditionally free Gaussian and conditionally free Poisson distribution.

arXiv comment26 pages (pictures from R. Speicher), AMS-TeX 3.0, HD-AM-BLS-01