On inverse moments of nonnegative random variables. (English) Zbl 0991.60003

The aim of this article is to investigate under which conditions the inverse moments of the form \(E[(1+X_n)^{-\alpha}]\) can be approximated by the inverse of the moments, i.e. expressions of the form \((1+EX_n)^{-\alpha}\), for a sequence of nonnegative random variables \((X_n)\), and a real number \(\alpha >0\). The authors provide (natural) sufficient conditions for which this is true, more precisely, that: \[ \lim_{n\to\infty} {E\bigl[ (1+X_n)^{-\alpha} \bigr] \over(1+EX_n)^{-\alpha}} =1. \]


60E05 Probability distributions: general theory
62E20 Asymptotic distribution theory in statistics
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