Let X, X1, X2, ... be i.i.d. S(n) = SIGMA1(n)X(j), E\X\ > 0, E(X) = 0 and tau = inf{n greater-than-or-equal-to 1 : S(n) greater-than-or-equal-to 0}. By Wald's equation, E(tau) = infinity. If E(X2) < infinity, then by a theorem of Burkholder and Gundy (1970), E(tau1/2) = infinity. In this paper, we prove that if E((X-)2) < infinity, then E(tau1/2) = infinity. When X is integer-valued and X greater-than-or-equal-to -1 a.s., a necessary and sufficient condition for E(tau1-1/p) < infinity, p > 1, is SIGMAn-1-1/P E\S(n)\ < infinity.
