A Note on the Inverse Moment for the Non Negative Random Variables

Abstract

Let {Z_n} be a sequence of non negative random variables satisfying a Rosenthal-type inequality and , where {M_n} is a sequence of positive real numbers. By using the Rosenthal-type inequality, the inverse moment E(a + X_n)^{− α} can be asymptotically approximated by (a + EX_n)^{− α} for all a > 0 and α > 0. Furthermore, we show that E[f(X_n)]^{− 1} can be asymptotically approximated by [f(EX_n)]^{− 1} for a function f( · ) satisfying certain conditions. Our results generalize and improve some corresponding results, which can allow immediate applications to compute the inverse moments for the non negative random variables whose distributions are such as Binomial distribution, Poisson distribution, Gamma distribution, etc.

Keywords:

• Rosenthal-type inequality
• Inverse moment
• Binomial distribution

Mathematics Subject Classification:

• 60E15
• 62E20
• 62G20