Limit Theorems for Monotonic Convolution and the Chernoff Product Formula

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 noncommutativity of this convolution, we use a new technique based on the Chernoff product formula. We also study this correspondence for multiplicative monotone convolution, where the Bercovici–Pata bijection no longer holds.