On the moments of some first passage times for sums of dependent random variables

Let Sn,n = 1, 2, ..., denote the partial sums of integrable random variables. No assumptions about independence are made. Conditions for the finiteness of the moments of the first passage times N(c) = min {n: Sn>ca(n)}, where c >= 0 and a(y) is a positive continuous function on [0, [infinity]), such that a(y) = o(y) as y --> [infinity], are given. With the further assumption that a(y) = yP, 0 <= p < 1, a law of large numbers and the asymptotic behaviour of the moments when c --> [infinity] are obtained. The corresponding stopped sums are also studied.