Asymptotic approximation of inverse moments of nonnegative random variables. (English) Zbl 1168.60340

Summary: Let \(\{Z_n,n\geq 1\}\) 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 \(\alpha \)th inverse moment \(E\{a+X_n\}^{ - \alpha} \) can be asymptotically approximated by the inverse of the \(\alpha \)th moment \(\{a+EX_n\} ^{- \alpha} \) where \(a>0,\alpha>0\), and \(\{X_n\}\) are the naturally-scaled partial sums. Furthermore, it is shown that, when \(\{Z_n\}\) only possess finite \(r\)th moments, \(1\leq r<2\), the preceding asymptotic approximation can still be valid by using different norming constants which are the standard deviations of partial sums of suitably truncated \(\{Z_n\}\).


60F99 Limit theorems in probability theory
62G20 Asymptotic properties of nonparametric inference
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