Conditionally monotone independence and the associated products of graphs

Abstract

We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence on suitably constructed larger algebras. For that purpose, we use the approach developed for a reduction of similar type for boolean, free and monotone independences. We apply the tensor product realization of c-monotone random variables to introduce the c-comb (loop) product of birooted graphs, a generalization of the comb (loop) product of rooted graphs, and we show that it is related to the c-monotone additive (multiplicative) convolution of distributions.