A note on asymptotic approximations of inverse moments of nonnegative random variables

Let {Zn}{Zn} be a sequence of independently distributed and nonnegative random variables and let Xn=∑i=1nZi. We show that, under mild conditions, E[(a+Xn)−α]E[(a+Xn)−α] can be asymptotically approximated by [a+E(Xn)]−α[a+E(Xn)]−α for a>0a>0 and α>0α>0. We further show that E{[f(Xn)]−1}E{[f(Xn)]−1} can be asymptotically approximated by {f[E(Xn)]}−1{f[E(Xn)]}−1 for a function f(⋅)f(⋅) satisfying certain conditions.