Probability Distributions on Bicompact Topological Groups

Abstract

In this paper random variables defined on a bicompact topological group G are examined. A regular completely additive measure $\mu $ defined on the class of all Borel subsets of G with $\mu (G) = 1$ will be called a probability distribution. Let $\xi _1 $, $\xi _2 $ be two independent G-valued random variables with probability distributions $\mu _1 $, $\mu _2 $. Then the product $\xi = \xi _1 \cdot \xi _2 $ has the probability distribution $\mu = \mu _1 \cdot \mu _2 $, where the convolution $\mu _1 \mu _2 $ is defined by the formula \[ {\text{(1)}}\qquad \mu _1 \mu _2 (E) = \int_G {\mu _1 (Ex^{ - 1} )d\mu _2 (x)} \] Let us denote by $S(G)$ the set of all probability distributions on a bicompact group G. By virtue of the multiplication defined in (1), the set $S(G)$ has become clearly a semigroup. It is possible to introduce into $S(G)$ a certain topology so that it will become a Hausdorff bicompact semigroup. The purpose of this paper is to find the class of all possible limiting distributions for a sequence of independent G-valued random variable compositions and the conditions for convergence. In each case the limit as $n \to \infty $ of $\mu ^n $, where, $\mu ^n $ is the n-th power of $\mu $, results in a stable distribution. (We say that a distribution is stable if the composition of two independent random variables with the given distribution also has the given distribution.) A distribution is stable, that is, idempotent if and only if it is the Haar measure of a closed subgroup H on G and 0 outside. The main result is Theorem 12: Let$\xi _{1} ,\xi _{2} , \cdots \xi _{n} \cdots $be a sequence of independent random variables. We construct the compositions\[ \xi _n = \xi _1 \xi _2 \cdots \xi _n .\]Then, non-random constants$a_n $can be found so that the sequence o f normalized compositions\[ \eta _n = \xi _n a_n \]has a limiting distribution. The results obtained are extended to the case of bicompact topological semigroups. Some material contained in this paper was published in [7] without detailed proofs.