Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.