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 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. 
Michael Anshelevich and John D. Williams
math.OA math.PR