Random walks on compact semigroups

Abstract:

Let $\beta$ be a regular Borel probability measure with support $F$ on a compact semigroup $S$. Let ${X_1},{X_2}, \cdots$ be a sequence of independent random variables on some probability space $(\Omega ,\Sigma ,P)$ with values in $S$, having identical distribution $P({X_n} \in B) = \beta (B)$. The random walk ${Z_n} = {X_1}{X_2} \cdots {X_n}$ is studied in this paper. Let $D$ be the closed semigroup generated by $F$. An element $x$ in $D$ is called recurrent iff ${P_x}({Z_n} \in {N_x}i.o.) = 1$ for every open set ${N_x}$ containing $x$. This paper characterizes the recurrence of an element $x$ in terms of divergence of the series $\sum \nolimits _{n = 1}^\infty {{\beta ^n}} ({N_x})$ for every open set ${N_x}$ containing $x$. It also shows that the set of recurrent states of $\{ {Z_n}\}$ is precisely the kernel of $D$.