Asymptotics for self-normalized random products of sums of i.i.d. random variables

AbstractLet {X,Xi;i⩾1} be a sequence of independent and identically distributed positive random variables, which is in the domain of attraction of the normal law, and tn be a positive, integer random variable. Denote Sn=∑i=1nXi, Vn2=∑i=1n(Xi−X¯)2, where X¯ denotes the sample mean. Then we show that the self-normalized random product of the partial sums, (∏k=1tnSkkμ)μVtn, is still asymptotically lognormal under a suitable condition about tn