Conditions of convergence of a random walk on a finite group
Tom 167 / 2022
Streszczenie
Let $P$ be a probability on a finite group $G$, and $V$ its carrier. Conditions of convergence of the $n$-fold convolution $P^{(n)}$ to the uniform probability on $G$ ($n\rightarrow \infty $) in terms of $V$ are known. We consider conditions under which the carrier $V^{n}$ of the probability $P^{(n)}$ converges as $n\rightarrow \infty $, i.e. $V^{n}=V^ {n+1}=\cdots $ for $n$ large. The convergence of $V^{n}$ is equivalent to the convergence of $P^{(n)}$. Instead of $G$ one can take its subgroup $\langle V \rangle $, generated by $V$. Then $V^{n}$ does not converge if and only if $V$ lies in a non-identity coset in $\langle V \rangle $ of a normal subgroup with a cyclic factor group. The article can be considered as a study of the behavior of the powers $V^{n}$ of a subset $V$ of a finite group as $n\rightarrow \infty $.
