Asymptotic approximation of inverse moments of nonnegative random variables

Let be a sequence of independent nonnegative r.v.'s (random variables) with finite second moments. It is shown that under a Lindeberg-type condition, the [alpha]th inverse moment E{a+Xn}-[alpha] can be asymptotically approximated by the inverse of the [alpha]th moment {a+EXn}-[alpha] where , and {Xn} are the naturally-scaled partial sums. Furthermore, it is shown that, when {Zn} only possess finite rth moments, 1<=r<2, the preceding asymptotic approximation can still be valid by using different norming constants which are the standard deviations of partial sums of suitably truncated {Zn}.