Limit Theorems and Wrapping Transforms in Bi-free Probability Theory
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.
Keywords
Cite
@article{arxiv.2007.02775,
title = {Limit Theorems and Wrapping Transforms in Bi-free Probability Theory},
author = {Takahiro Hasebe and Hao-Wei Huang},
journal= {arXiv preprint arXiv:2007.02775},
year = {2020}
}