A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, X n ; n . 1} be a sequence of i.i.d. Banach space valued random variables and let {a n ; n . 1} be a sequence of positive constants such that an..and1<liminfn..a2nan.limsupn..a2nan<.. Set Sn=.ni=1Xi,n.1. In this paper we prove that .n.11nP(.Sn...an)<.forall.>0 if and only if limn..Snan=0a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.