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arXiv · 2007.02775

Limit Theorems and Wrapping Transforms in Bi-free Probability Theory

Takahiro Hasebe·Hao-Wei Huang
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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.

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Takahiro Hasebe, Hao-Wei Huang. 2020-07-06. Limit Theorems and Wrapping Transforms in Bi-free Probability Theory. https://arxiv.org/abs/2007.02775

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