A universal result in almost sure central limit theory

The discovery of the almost sure central limit theorem (Brosamler, Math. Pr oc. Cambridge Philos. Sec. 104 (1988) 561-574; Schatte, Math. Nachr. 137 (1 988) 249-256) revealed a new phenomenon in classical central limit theory a nd has led to an extensive literature in the past decade. In particular, a. s. central limit theorems and various related 'logarithmic' limit theorems have been obtained for several classes of independent and dependent random variables. In this paper we extend this theory and show that not only the c entral limit theorem, but every weak limit theorem for independent random v ariables, subject to minor technical conditions, has an analogous almost su re version. For many classical limit theorems this involves logarithmic ave raging, as in the case of the CLT, but we need radically different averagin g processes for 'more sensitive' limit theorems. Several examples of such a .s. limit theorems are discussed. (C) 2001 Elsevier Science B.V. All rights reserved.