ON ITO-NISIO TYPE THEOREMS FOR DS-GROUPS

Abstract

It is shown that the convergence of convolution products of probability measures on certain non-locally compact topological abelian groups can be veriÞed by means of characteristic function- als. Analogous results are obtained also for almost everywhere con- vergence of series of independent random elements in the considered groups. A connection with the Sazonov property of the groups is discussed. 1. Preliminaries. Throughout the paper N denotes the set of natu- ral numbers; Q;R, and C are, respectively, the Þelds of rational, real, and complex numbers with the ordinary (Euclidean) metric, and T denotes the multiplicative group of complex numbers of modulus 1 with the metric in- duced from C. For a topological abelian group X we denote by X 0 the topological dual group which consists of all continuous (unitary) characters h : X !T; the group operation in X 0 is the natural pointwise multiplication. No topology in X 0 is speciÞed. A topological abelian group X is called dually separated or DS-group if X 0 separates the points of X; in other words, X is a DS-group if for any dierent x1;x2 2 X there is a character h 2 X 0 such that h(x1) 6 h(x2). Hausdorlocally compact abelian (LCA-) groups and any additive subgroup of any Hausdorlocally convex space are examples of DS-groups. Below we shall see another type of examples too. Let X be a completely regular Hausdortopological space. Denote by Mt(X) the set of all Radon probability measures n deÞned on the Borel o-algebra of X. In Mt(X) we consider only the weak topology (for all the notions unexplained here the reader is referred to (1) and (2)). For Þxed x 2 X we denote by ex the Dirac measure concentrated at x. The Prokhorov theorem says that a subset M o Mt(X) is relatively compact if