  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.
