Compound Bi-free Poisson Distributions

TL;DR

This paper investigates compound bi-free Poisson distributions for two-faced random variables, establishing limit theorems, infinite divisibility, and constructing matrix models under various conditions.

Contribution

It introduces new limit theorems and models for compound bi-free Poisson distributions, expanding the understanding of their structure and representations.

Findings

Proves a Poisson limit theorem for compound bi-free distributions.

Shows infinite divisibility as limits of compound bi-free Poisson distributions.

Constructs random bi-matrix models under specific conditions.

Abstract

In this paper, we study compound bi-free Poisson distributions for {\sl two-faced families of random variables}. We prove a Poisson limit theorem for compound bi-free Poisson distributions. Furthermore, a bi-free infinitely divisible distribution for a two-faced family of self-adjoint random variables can be realized as the limit of a sequence of compound bi-free Poisson distributions of two-faced families of self-adjoint random variables. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a two faced family of finitely many random variables, which has an almost sure random matrix model, and the left random variables commute with the right random variables in the two-faced family, then we can construct a random bi-matrix model for the compound bi-free Poisson distribution. If a compound bi-free Poisson distribution is determined by a positive number and the distribution of a commutative pair of random variables, we can construct an asymptotic bi-matrix model with entries of creation and annihilation operators for the compound bi-free Poisson distribution.