Central limit theorems of sums of powers of function of independent random variables

Abstract

Let (Xnk), k = 1,..., kn ; n = 1, 2,... be a double sequence of infinitesimal random variables which are rowwise independent, under a condition of convergence of distribution function of the sums Sn = Xn1 + ... + Xnkn - An where An are suitably chosen constants, we give necessary and sufficient conditions for the sequence of distribution functions of Srn = (g(Xn1))r + ... + (g(Xnkn))r - Bn(r) to weakly converge to a limiting distribution function Fr for each natural number r, and also for convergence of the sequence of distribution function (Fr), where g:R-->R has certain properties and Bn(r) are suitably chosen constants.