Multidimensional Compound Poisson Distributions in Free Probability

TL;DR

This paper establishes an infinite dimensional compound Poisson limit theorem within free probability, introduces related distributions, and characterizes infinite divisibility through free cumulants and limit behaviors.

Contribution

It extends free probability theory by defining and characterizing infinite dimensional compound free Poisson distributions and proving a new limit theorem.

Findings

Proved an infinite dimensional compound Poisson limit theorem.

Defined and characterized infinite dimensional free infinitely divisible distributions.

Established equivalences for multidimensional free infinite divisibility.

Abstract

Inspired by R. Speicher's multidimensional free central limit theorem and semicircle families, we prove an infinite dimensional compound Poisson limit theorem in free probability, and define infinite dimensional compound free Poisson distributions in a non-commutative probability space. Infinite dimensional free infinitely divisible distributions are defined and characterized in terms of its free cumulants. It is proved that for a distribution of a sequence of random variables, the following statements are equivalent. (1) The distribution is multidimensional free infinitely divisible. (2) The distribution is the limit distribution of triangular trays of families of random variables. (3) The distribution is the distribution of $\{a_1^{(i)}: i=1, 2, \cdots\}$ of a multidimensional free Levy process $\{\{a_t^{(i)}:i=1, 2, \cdots\}: t\ge 0\}$. (4) The distribution is the limit distribution of a sequence of multidimensional compound free Poisson distributions.