Generalized binomial law and regularly varying moments

Abstract

In this paper we demonstrate a method for estimating asymptotic behavior of the regularly varying moments $E(K_\rho (X_n))$, $(n\toınfty)$ in the case of generalized Binomial Law. Here $K_\rho(x)$ is from the class of regularly varying functions in the sense of Karamata.We prove that $$E(K_\rho(X_n))\sim K_\rho(E(X_n)), \ \rho>0, \ \ \ E(X_n)\toınfty \ \ \ (n\toınfty),$$i.e., that the asymptotics of the first moment determines thebehavior of all other moments.