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A note on asymptotic approximations of inverse moments of nonnegative random variables. (English) Zbl 1191.62020

Summary: Let \(\{Z_n\}\) be a sequence of independently distributed and nonnegative random variables and let \(X_n=\sum^n_{i=1}Z_i\). We show that, under mild conditions, \(E[(a+X_n)^{-\alpha}]\) can be asymptotically approximated by \([a+E(X_n)]^{-\alpha}\) for \(a>0\) and \(\alpha >0\). We further show that \(E\{[f(X_n)]^{-1}\}\) can be asymptotically approximated by \(\{f[E(X_n)]\}^{-1}\) for a function \(f(\cdot)\) satisfying certain conditions.

MSC:

62E20 Asymptotic distribution theory in statistics
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