On the universal AS central limit theorem

Let (X k ) be a sequence of independent r.v.'s such that for some measurable functions gk : Rk → R a weak limit theorem of the form gk (X1 , ... ,Xk)→ L G holds with some distribution function G. By a general result of Berkes and Csáki ("universal ASCLT"), under mild technical conditions the strong analogue 1/DN ∑k = 1Ndk I{g k (X1, ... ,Xk) ≦ x} → G(x)} a.s. is also valid, where (d k ) is a logarithmic weight sequence and D N = ∼ k=1N d k . In this paper we extend the last result for a very large class of weight sequences (d k ), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory, are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with, ordinary averages. © 2007 Springer Science + Business Media B.V.