Convolution and Limit Theorems for Conditionally Free Random Variables

TL;DR

This paper introduces the concept of conditionally free convolution, exploring its combinatorial and analytic properties, and explicitly deriving distributions for specific cases like Gaussian and Poisson.

Contribution

It presents the first detailed study of conditionally free convolution, connecting it with non-crossing partitions and providing explicit distribution formulas.

Findings

Defined conditionally free convolution and product

Connected convolution with non-crossing partitions

Derived explicit distributions for specific cases

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.