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

Abstract

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