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

Let {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 = n i=1 Xi , V 2 n = n i=1(Xi − X)2, where X denotes the sample mean. Then we show that the selfnormalized random product of the partial sums, ( tn k=1 Sk kμ ) μ Vtn , is still asymptotically lognormal under a suitable condition about tn. © 2007 Elsevier Inc. All rights reserved