Conditions for the limit distribution equiprobability in a linear autoregression scheme with random control on a finite group

We consider the sequence of random variables
$$ \mu^{(N)}=\xi_N(\mu^{(N-1)})^{\zeta_N}, \qquad N=1,2,\dots, $$
where $\mu^{(0)}$ is a random variable that takes values in a finite group $G=(G, \bullet)$, $(\xi_N, \zeta_N)$, $N=1,2,\dots$, is a sequence of identically distributed random variables that take values in the Cartesian product $G\times\operatorname{Aut}G$, where $(\operatorname{Aut}G, \circ)$ is the group of automorphisms of $G$. We assume that the random variables $\mu^{(0)}$, $(\xi_N, \zeta_N)$, $N=1,2,\dots$, are independent. Given an arbitrary distribution of $\mu^{(0)}$, we find general necessary and sufficient conditions for the convergence, as $N\to\infty$, of the sequence of distributions of random variables $\mu^{(N)}$ to the equiprobable on $G$ distribution.
This research was supported by the Program of the President of the Russian Federation for supporting the leading scientific schools, grant 2358.2003.9.