Convergence of Convolution Sequences of Probability Measures

Abstract

The study of the convergence of sequences of probability measures on a locally compact group is basic for the development of the central limit theorem. This chapter serves as a source of preparatory material as well as a presentation of the smooth part of the theory. Methodically the problems treated are chosen from the axiomatic point of view: Classical theorems are not only generalized to certain sufficiently large classes of locally compact groups, but in addition their domains of validity are determined. The fact that convergence of convolution sequences is a rare event is indicated in the first section. The analysis of conditions for the convergence of the sequence of powers of a probability measure makes evident all the difficulties which can arise in solving the general problem. The theorem of Ito and Kawada and its limitations are the main result in this direction. The theorem is valid only in the special case of a compact group. The next problem solved is the equivalence of stochastic, a.s. convergence and convergence in distribution for sequences of n-fold products of group-valued random variables. The domain of validity of this equivalence principle appears to be the class of locally compact groups without nontrivial compact subgroups. In a similar way we discuss the convergence principle in its various forms and analyze the convergence behavior of a sequence of convolution products after a certain shift. According to the particular shift one abtains probabilistic characterizations of the classes of all compact groups and of all totally disconnected compact groups.