Limit Theorems for Convolution Iterates of a Probability Measure on Completely Simple or Compact Semigroups

Abstract

Abstract. This paper extends the study (initiated by M. Rosenblatt) of the asymptotic behavior of the convolution sequence of a probability measure on compact or completely simple semigroups. Let S be a locally compact second countable Hausdorff topological semigroup. Let ji be a regular probability measure on the Borel subsets of S such that S does not have a proper closed subsemigroup containing the support F of /x. It is shown in this paper that when S is completely simple with its usual product representation X X G X Y, then the convolution sequence ¡i" converges to zero vaguely if and only if the group factor G is noncompact. When the group factor G is compact, ¡i" converges weakly if and only if limn_>00 F" is nonempty. This last result remains true for an arbitrary compact semigroup S generated by F. Furthermore, we show that in this case there exist elements an G S such that ¡i" « 5ĉ onverges weakly, where 5a> is the point mass at an. This result cannot be extended to the locally compact case, even when 5 is a group.1. Throughout this paper, 5 will denote a locally compact second countable semigroup (i.e. an algebraic semigroup with locally compact Hausdorff topology and jointly continuous multiplication).By a measure on S, we will mean a finite regular nonnegative measure on the class of all Borel sets (generated by open sets) of S. Let C(S) be the realvalued continuous functions with compact support. A net of measures (ßk) is said to converge in the weak*-sense (or vaguely) to a measure ß if and only if for each/ E C(S), ffdßk converges to ffdß. By Banach-Alaoglu's theorem, the set B(S) of all measures ß with ß(S) < 1 is compact in the weak*-topology. Also B(S) is an algebraic semigroup under the usual convolution (*) operation of measures.