Limit theorems for monotonic convolution and the Chernoff product formula

TL;DR

This paper extends the Bercovici-Pata correspondence of limit theorems to monotone convolution, using the Chernoff product formula to handle non-commutativity and exploring both additive and multiplicative cases.

Contribution

It introduces a new technique based on the Chernoff product formula to extend limit theorem correspondences to monotone convolution, including non-commutative cases.

Findings

Extended the Bercovici-Pata correspondence to additive monotone convolution.

Developed a new technique using the Chernoff product formula for non-commutative convolutions.

Analyzed the limits for multiplicative monotone convolution where the bijection does not hold.

Abstract

Bercovici and Pata showed that the correspondence between classically, freely, and Boolean infinitely divisible distributions holds on the level of limit theorems. We extend this correspondence also to distributions infinitely divisible with respect to the additive monotone convolution. Because of non-commutativity of this convolution, we use a new technique based on the Chernoff product formula. In fact, the correspondence between the Boolean and monotone limit theorems extends from probability measures to positive measures of total weight at most one. Finally, we study this correspondence for multiplicative monotone convolution, where the Bercovici-Pata bijection no longer holds.