On the Relationship Between the Baum.Katz.Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {X n ; n . 1} and a sequence of positive constants {a n ; n . 1}, the relationship between the Baum.Katz.Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which (i) limsupn...Sn.an<.a.s. and .n=1.1nP(.Sn.an..)<.forall.>.forsomeconstant ..[0,.) are equivalent; (ii) For all constants . . [0,.), limsupn...Sn.an=.a.s. and .n=1.1nP(.Sn.an..){<., = .,if.>.if.<. are equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.