CONVERGNECE RATES IN THE LAW OF LARGE NUMBERS FOR B-VALUED RANDOM ELEMENTS 1

Abstract The author discusses necessary and sufficient conditions of the complete con-vergence for sums of B-valued independent but not necessarily identically distributed r.v.′s in Banach space of type p, and obtain characterization of Banach space of type p in terms of the complete convergence. A series of classical results on iid real valued r.v.′s are extended. As application author gives the analogous results for randomly indexed sums. Key words Convergence rate, random element, Banach space of type p, slowly varying function 1991 MR Subject Classification 60B12 1 Introduction and Main Results Let {Xn, n ≥ 1} be a sequence of random variables in the same probability space and put Sn = n∑ i=