Recently, Bercovici has introduced multiplicative convolutions based on
Muraki's monotone independence and shown that these convolution of probability
measures correspond to the composition of some function of their Cauchy
transforms. We provide a new proof of this fact based on the combinatorics of
moments. We also give a new characterisation of the probability measures that
can be embedded into continuous monotone convolution semigroups of probability
measures on the unit circle and briefly discuss a relation to Galton-Watson
processes.Comment: 14 page