On convergence rates and the expectation of sums of random variables

Let Xi be iidrv's and Sn=X1+X2+...+Xn. When EX21<+[infinity], by the law of the iterated logarithm (Sn-[alpha]n)/(n log n)1/2-->0 a.s. for some constants [alpha]n. Thus the r.v. Y=supn[greater-or-equal, slanted]1[Sn-[alpha]n-([delta]n log n)1/2]+ is a.s.finite when [delta]>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX21<+[infinity] that EYh<+[infinity] if and only if EX12+h[logX1]-1<+[infinity] for 0<h<1 and [delta]> hE(X1-EX1)2, whereas EYh=+[infinity] whenever h>0 and 0<[delta]<hE(X1-EX1)2.