Conditionally monotone independence and the associated products of graphs

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.

Convolution of distributions in sequential approach

A new class of special upper approximate units together with the known Vladimirov class of special approximate units are used to consider various sequential conditions of convolvability of distributions. The equivalence of these conditions to the known conditions of convolvability given by C. Chevalley and L. Schwartz is proved together with the equivalence of the corresponding definitions of the convolution of distributions.