Z. Wahrscheinlichkeitstheorie verw. Gebiete 36, 1 - 7 (1976) Zeitschrift f~ir Wahrscheinlichkeitstheorie und verwandteGebiete by Springer-Verlag 1976 A Convergence of Types Theorem for Probability Measures on Topological Vector Spaces with Applications to Stable Laws I. Csiszfir 1 and Balram S. Rajput 2 1 Mathematical Institute of the Hungarian Academy of Sciences, 1053 Budapest, ReAltanoda u. 13-15, Hungary 2 Department of Mathematics, University of Tennessee, Knoxville, Tenn. 37916, USA 1. Introduction The most frequently used tool for the purpose of characterization of stable laws on real, separable Hilbert and Banach spaces is an appropriate version of the convergence of types theorem for measures in these spaces (see, for example, Breiman [1], Jajte [5] and Kumar and Mandrekar [6]). This paper is devoted to a generalization of the convergence of types theorem for probability measures on arbitrary TVS (Theorem 1). We also demonstrate the equivalence of the three standard characterizations of stable laws in this general TVS setting (Theorem 2). Since Fourier analysis and Pro~orov's compactness theorem are not available, other techniques are needed; we shall rely on those developed in [2] and [3] for arbitrary topological groups. All results below are valid under the hypothesis of K-regularity for the measures considered but some will be proved under the weaker hypothesis of z-regularity, the weakest one under which-to our knowl- edge-convolution of measures has been meaningfully defined. On most TVS of interest, all probability measures are z-regular, cf. Fernique [-4, pp. 11 and 17]. 2. Preliminaries Unless stated otherwise, the following notations and conventions will remain fixed throughout the paper. Some definitions and known facts needed in the sequel are also listed. P. 1. By a measure on a topological space we shall always mean a probability measure defined on the Borel o--algebra of the space (the smallest a-algebra containing the open sets). A measure # is K-regular if #(A)=suPA#(K ) for every Borel set A where K ranges over the compact subsets of A; # is z-regular if #(G)= lim #(G~) for every increasing net of open sets G~TG. P.2. IR and IR + will denote the reals and the positive reals, respectively. IE will denote a Hausdorff TVS (topological vector space) over 1R, and 0 its zero element.