An extension of almost sure central limit theory

We show that if X-n are i.i.d. random variables with EX1 = 0, EX12 = 1 and (d(k)) is a positive numerical sequence obeying a condition similar to Kolmogorov's condition for the LIL, then letting D-n = Sigma(n)(k=1) d(k) we have [GRAPHICS] This shows that logarithmic means, used traditionally in a.s. central limit theory, can be replaced by other means lying much closer to ordinary (Cesaro) averages, leading to considerably sharper results. Our results remain valid (under suitable regularity conditions) for independent random variables X, satisfying the weak limit theorem a(n)(-1) S-n - b(n) (L)under right arrow H with an arbitrary distribution function H. (C) 2005 Elsevier B.V. All rights reserved.