Let $S_n,n$ = 1, 2,... denote the partial sums of not necessarily in-dependent random variables. Let N(c) = min${ n ; S_n > c}$, c $geq$ 0. Theorem 2 states that N (c), (suitably normalized), tends to 0 in p-mean, 1 $leq$ p < 2, as c longrightarrow $infty$ under mild conditions, which generalizes earlier result by Gut(1974). The proof follows by applying Theorem 1, which generalizes the known result $E$mid$S_n$mid$^p$</TEX> = o(n), 0 < p< 2, as n .rarw..inf. to randomly indexed partial sums.