Necessary and sufficient convergence conditions for the convolution of non-identical distributions given on a finite group

Say that the convolution of a sequence of distributions $x_1,x_2,\dots,x_n,\dots$ on a finite group $G$ converges if, for all $i=1,2,\dots,$ the sequences $x_ix_{i+1}\dots x_{i+n}$ converge as $n\to\infty$, each $x_i$ being viewed as the element ${p_1}^ie_1+\dots+{p_s}^ie_s$ of the algebra over the field of real numbers with the basis $e_1,\dots,e_s\in G$, where ${p_k}^i$ is the probability of $e_k$ given by $x_i$.
In the paper the necessary and sufficient conditions of such a convergence are found. In particular, the necessary and sufficient conditions are obtained that $\{x_i\dots x_{i+n}\}$, $i=1,2,\dots$, converge to the uniform distribution on $G$.