Generalized binomial law and regularly varying moments

Abstract

In this paper we demonstrate a method for estimating asymptotic behavior of the regularly varying moments $E(K_{\rho }(X_n))$, $(n\to ınfty)$ 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 $$E(K_{\rho }(X_n))\sim K_{\rho }(E(X_n)),\rho >0,E(X_n)\to ınfty(n\to ınfty),$$i.e., that the asymptotics of the first moment determines thebehavior of all other moments.