Asymptotics for self-normalized random products of sums for mixing sequences

Let {X, X-n, n >= 1} be a sequence of a strictly stationary phi-mixing positive random variables, which is in the domain of attraction of the normal law, and t(n) be a positive, integer random variable and denote S-n = Sigma(n)(i=1) X-i, V-n(2) = Sigma(n)(i=1) X-i(2), and EX = mu > 0. Under a general condition about t(n) and Sigma(infinity)(i=1) phi(1/2) (i) < infinity, we show that the self-normalized random products of the partial sums, (Pi(tn)(j=1) S-k/k mu)V-tn/mu, is still asymptotically lognormal.