On inverse moments of nonnegative random variables (Q5952089)

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: NEWLINE\[NEWLINE\lim_{n\to\infty} {E\bigl[ (1+X_n)^{-\alpha} \bigr] \over(1+EX_n)^{-\alpha}} =1.NEWLINE\]