Asymptotic approximation of inverse moments of nonnegative random variables

Let {Zn,n≥1} 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 ααth inverse moment E{a+Xn}−αE{a+Xn}−α can be asymptotically approximated by the inverse of the ααth moment {a+EXn}−α{a+EXn}−α where a>0,α>0, and {Xn}{Xn} are the naturally-scaled partial sums. Furthermore, it is shown that, when {Zn}{Zn} only possess finite rrth moments, 1≤r<21≤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}{Zn}.