LIMIT THEOREMS AND WRAPPING TRANSFORMS IN BI-FREE PROBABILITY THEORY

We characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus. The bi-free analogous L & eacute;vy 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 measures induced by the wrapping map from the plane to the bi-torus.