Asymptotic approximation of inverse moments of nonnegative random variables

Abstract

Let { Z n , 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 α th inverse moment E { a + X n } − α can be asymptotically approximated by the inverse of the α th moment { a + E X n } − α where a > 0 , α > 0 , and { X n } are the naturally-scaled partial sums. Furthermore, it is shown that, when { Z n } only possess finite r th 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 { Z n } .