## 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:

 6.2e+21 Asymptotic distribution theory in statistics
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### References:

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