On the almost sure central limit theorem and domains of attraction (Q1892255)

A sequence \((X_ n)\) of i.i.d. r.v.'s with distribution function \(F\) and \(S_ n = X_ 1 + \cdots + X_ n\) is said to satisfy the a.s. central limit theorem (ASCLT) if there exist numerical sequences \((a_ n)\), \((b_ n)\) such that \[ \lim_{N \to \infty} {1 \over \log N} \sum_{k \leq N} {1 \over k} I \left\{ {S_ k \over a_ k} - b_ k < x \right\} = \Phi (x) \text{ a.s. for all } x \] holds, where \(I\) denotes the indicator function. In a recent paper, the author and \textit{H. Dehling} [J. Theor. Probab. 7, No. 3, 667-680 (1994; Zbl 0804.60021)] gave the sufficient condition \(\lim_{x \to \infty} f(x) = 0\) for the ASCLT, where \(f(x) = x^ 2 (1 - F(x) + F( - x))/ \int_{| t | \leq x} t^ 2 dF(t)\), and the necessary condition \(\liminf_{x \to \infty} f(x) = 0\). The purpose of the paper is to close the gap between these conditions and to find a necessary and sufficient condition for the ASCLT. The method leads also to new results on the limit distributional behavior of \(S_ n/a_ n - b_ n\) along subsequences (partial attraction), as well as to necessary and sufficient conditions for averaged versions of the central limit theorem \[ \lim_{N \to \infty} {1 \over Q_ N} \sum_{k \leq N} q_ k P \left\{ {S_ k \over a_ k} - b_ k < x \right\} = \Phi (x) \text{ a.s. for all }x, \] where \((q_ n)\) is a sequence of positive weights with \(Q_ N = \sum_{k \leq N} q_ k \to \infty\) as \(N \to \infty\).