On the asymptotic approximation of inverse moment for non negative random variables

In this paper, we have studied the asymptotic approximation of inverse moment. Let {Zn} be non negative random variables, where the truncated random variables satisfy the Rosenthal-type inequality. Denote X˜n=∑i=1nZi$\tilde{X}_n=\sum \nolimits _{i=1}^n Z_i$. The inverse moment E(a+X˜n)-α$E(a+ \tilde{X}_n)^{-\alpha }$ can be asymptotically approximated by (a+EX˜n)-α$(a+E\tilde{X}_n)^{-\alpha }$, and the growth rate is presented as |E(a+X˜n)-α/(a+EX˜n)-α-1|=O(n/EX˜n)$|E(a+\tilde{X}_n)^{-\alpha }/(a+E\tilde{X}_n)^{-\alpha }-1|=O(\sqrt{n}/E\tilde{X}_n)$. Meanwhile, we obtain a result E[f(X˜n)]-1∼[f(EX˜n)]-1$E[f(\tilde{X}_n)]^{-1}\sim [f(E\tilde{X}_n)]^{-1}$ for a function f(x) satisfying certain conditions. On the other hand, if Xn=∑i=1nZi/∑i=1nVar(Zi)$X_n=\sum \nolimits _{i=1}^n Z_i/\sqrt{\sum \nolimits _{i=1}^n \hbox{\rm Var}(Z_i)}$, then the growth rate is presented as |E(a + Xn)− α/(a + EXn)− α − 1| = O(1/EXn). Our results generalize and improve some corresponding ones. Finally, some examples of inverse moment are illustrated.