A limit theorem related to the Hartman–Wintner–Strassen LIL and the Chover LIL

Let {X,Xn;n≥1} be a sequence of i.i.d. real-valued random variables, and let Sn=∑i=1nXi,n≥1. Write logx=loge(e∨x)logx=loge(e∨x), x≥0x≥0. In this note we establish a limit theorem which is related to the classical Hartman–Wintner–Strassen law of the iterated logarithm and the classical Chover law of the iterated logarithm. That is, for 1≤p<∞1≤p<∞, we show that lim supn→∞|Snn|(logloglogn)−1=ep/2almost surely if and only if EX=0andinf{b>0:limx→∞(loglogx)1−bE(X2I(|X|≤x))=0}=p.