Convolution products of probability measures on completely simple semigroups
Convolution products of probability measures are considered in the context of completely simple semigroups. Given a sequence of measures \((\mu_n)\subset\text{Prob}(S)\), where \(S\) is a finite completely simple semigroup, define the convolution products \[ \mu_{k,n} = \mu_{k+1}*\mu_{k+2}*\cdots \mu_n. \] The aim of the paper is to determine under what conditions \(\mu_{k_n}\) will converge weakly, as \(n\to\infty\), for all \(k > 0\). Results are proven which (1) relate the supports of the measures \(\mu_n\) to the supports of the tail limit measures, introduced by \textit{I. Csiszár} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 5, 279--295 (1966; Zbl 0144.39504)] and (2) determine necessary and sufficient conditions for convergence of the convolution products in the case of rectangular groups, i. e., completely simple semigroups where the Rees product structure is a direct product. An example showing how the theory can be used to analyze the convergence behavior of nonhomogeneous Markov chains is included.
scientific article; zbMATH DE number 5017410