Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Abstract

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 Φ, that is, lim F n (x)= Φ(x), as n → ℞, uniformly in x ∈ ℝ. We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx −α−1 lnγ(x), x > r, where α ⩾ 2, γ ∈ ℝ, 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 −β/2 lnγ n, n −β/2 lnγ-1 n, etc., where β ⩾ 0.