Nonlinear Analysis, Theory, Methods &Applications. Vol. 30, No. 7, pp. 42374248, 1997 Proc. 21ld World Congress of Nonlinear Analysts Q 1997 Elsevier Science Ltd PII: SO362-546X(96)00217-9 Rimed in Great Britain. All tights reserved 0362-546X/97 $17.00+0.00 ON THE RATE OF CONVERGENCE OF SERIES OF BANACH SPACE VALUED RANDOM ELEMENTS ANDREW ROSALSKYt and JOSEPH ROSENBLATTSS tDepartment of Statistics, University of Florida, Gainesville, Florida 32611, U.S.A.; SDepartment of Mathematics, UniversitTtf Illinois, Urbana, Illinois 61801, U.S.A. Key words and phrases: Real separable Banach space, Rademacher type p Banach space, convergent series of random elements, tail series, independent random elements, almost sure convergence, convergence in probability. 1. INTRODUCTION Throughout, {V,, n > 1) is a sequence of random elements defined on a probability space (a, 3, P) and taking values in a real separable Banach space X with norm 11 . I). As usual, their partial sums will be denoted by S, = Cy=, Vj, n 2 1. If S,, converges almost surely (a.s.) to a random element S, then (set SO = 0) T,rS-S m n-l = c Vj,n 2 1 j=n is a well-defined sequenceof random elements (referred to as the tail series) with T,, -+ 0 as. (1.1) In this paper, we shall be concerned with the rate in which S, converges to S or, equivalently, in which the tail seriesT,, converges to 0. More specifically, recalling that (1.1) is equivalent to sup IlTkll -r; 0, k>n we will provide conditions for the limit law SUPk>n IlTkll P+ o bn (1.2) to hold where {b,, n 2 1) is a sequenceof positive constants. These results are, of course, of greatest interest when b, = o(l). SResearch supported in part by NSF Grant DMS-94-01093. 4237