"A universal result in almost sure central limit theory"
István Berkes and Endre Csáki
The discovery of the almost sure central limit theorem (Brosamler,
1988; Schatte, 1988) revealed a new phenomenon in classical central limit
theory and 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 central limit theorem, but every weak limit
theorem for independent random variables, subject to minor technical
conditions, has an analogous almost sure version. For many classical
limit theorems this involves logarithmic averaging, as in the case of the
CLT, but we need radically different averaging processes for `more
sensitive' limit theorems. Several examples of such a.s. limit theorems
Keywords: Almost sure central limit theorem, logarithmic averages,
2000 Mathematics Subject Classification: 60F15;