On convergence rates and the expectation of sums of random variables

AbstractLet Xi be iidrv's and Sn=X1+X2+…+Xn. When EX21<+∞, by the law of the iterated logarithm (Sn−αn)(n log n)12→0 a.s. for some constants αn. Thus the r.v. Y=supn⩾1[|Sn−αn|−(δn log n)12]+ is a.s.finite when δ>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<+∞ that EYh<+∞ if and only if E|X1|2+h[log|X1|]-1<+∞ for 0<h<1 and δ> hE(X1−EX1)2, whereas EYh=+∞ whenever h>0 and 0<δ