Two-faced Families of Non-commutative Random Variables Having Bi-free Infinitely Divisible Distributions

TL;DR

This paper investigates the properties of two-faced families of non-commutative random variables with bi-free infinitely divisible distributions, establishing a limit theorem and characterizations via Levy processes.

Contribution

It introduces a limit theorem for sums of bi-free two-faced pairs and characterizes bi-free infinite divisibility using operator models and Levy processes.

Findings

Proves a limit theorem for bi-free two-faced pairs.

Shows equivalence between infinite divisibility and limit distributions.

Characterizes infinite divisibility via bi-free Levy processes.

Abstract

We study two-faced families of random variables having bi-free infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced pairs of random variables within a triangular array. Then, by using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.