Limit Theorems and Wrapping Transforms in Bi-free Probability Theory

TL;DR

This paper explores limit theorems and transforms in bi-free probability, characterizing idempotent distributions, deriving Levy triplets, and analyzing convergence criteria for multiple convolutions on the bi-torus.

Contribution

It provides a characterization of idempotent distributions and the Levy triplet in bi-free probability, establishing connections with classical probability and analyzing convergence conditions.

Findings

Unique bi-free Levy triplet for infinitely divisible distributions

Homomorphism between bi-free and classical multiplicative semigroups

Conditions for uniqueness of classical multiplicative Levy triplet

Abstract

In this paper, we characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus. Also, the bi-free analogous Levy triplet of an infinitely divisible distribution on the bi-torus without non-trivial idempotent factors is obtained. This triplet is unique and generates a homomorphism from the bi-free multiplicative semigroup of infinitely divisible distributions to the classical one. The relevances of the limit theorems associated with four convolutions, classical and bi-free additive convolutions and classical and bi-free multiplicative convolutions, are analyzed. The analysis relies on the convergence criteria for limit theorems and the use of push-forward measures induced by the wrapping map from the plane to the bi-torus. Different from the bi-free circumstance, the classical multiplicative L\'{e}vy triplet is not always unique. Due to this, some conditions are furnished to ensure uniqueness.