Strong uniform convergence of composition sequences of probability measures on locally compact topological semigroups (Q1368222)

Let \(S\) be a locally compact topological semigroup and \(\{\mu_n,\;n=1,2,\dots\}\) a sequence of probability measures on \(S\). Consider the convolution \(\mu_{k+1}*\mu_{k+2}*\cdots* \mu_n\) by \(\mu_{k,n}\) and assume further that \(\{\mu_{k,n},\;k=1,2,\dots,k< n\}\) is a tight set of probability measures. The sequence \(\{\mu_n\}\) is said to be composition convergent if \(\forall k:\mu_{k,n}\to \lambda_k\) weakly, as \(n\to\infty\), where \(\lambda_k\) is a probability measure on \(S\). The author investigates the consequences of composition convergence in different semigroups. In particular, he establishes a number of cases where the sequence \(\lambda_k\) converges. The limit is then necessarily the Haar measure on some compact subgroup \(H\) of \(S\). The key tool is the analysis of the relationship between a convolution semigroup \(N\) of probability measures on \(S\) and its support set \(\text{supp }N\). The results are closely related to those of Theorem 1 in [\textit{G. Budzban} and \textit{A. Mukherjea}, J. Theor. Probab. 5, 283-307 (1992; Zbl 0758.60007)]. \{The reviewer does not understand the proof of the assertion that the support sets \(G_1\) (in Corollaries 1 and 2) and \(S_1\) (in Corollary 3) are closed subsets of \(S\)\}.