Abstract
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We characterize idempotent distributions with respect to the bi-free multiplicative
convolution on the bi-torus. The bi-free analogous Lévy triplet of an infinitely
divisible distribution on the bi-torus without nontrivial 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. Also,
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.
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Keywords
infinite divisibility, multiplicative convolution, wrapping
transformation
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Mathematical Subject Classification
Primary: 46L54
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Milestones
Received: 7 December 2023
Revised: 5 April 2024
Accepted: 5 April 2024
Published: 12 June 2024
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| © 2024 MSP (Mathematical Sciences
Publishers). Distributed under the Creative Commons
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