Abstract. The main result of this paper can be formulated as follows: Given a sequence of real-valued random variables $\Phi = (\Phi_n)$ on a probability space $\Omega $ and given a sequence $c = (c_n)$ of reals, then there exists a subsequence $\Phi ' = (\Phi_{k_n})$ of $\Phi $ such that $$\lim_{N \to\infty } F_{W_N} (x) = {1\over\sqrt {2 \pi }}\int_{- \infty }^x e^{- t^2\over2} dt \hbox{ for } x\in{\msbm R}, W_N = {1\over C_N}\sum_{n=1}^N c_n \Phi_{k_n},$$ where $F_{W_N} (x)$ is the distribution function of the weighted sum $W_N$, provided that the following conditions are satisfied: $$C_N^2=\sum_{n=1}^N c_n^2\to\infty, c_N = o(C_N), \Phi_n\in L^2$$ with $\Phi_n\to0$ weakly in $L^1$ and $\Phi_n^2\to1$ weakly in $L^1$. Moreover, the subsequence $\Phi '$ may be chosen independently of the sequence $c$. A corresponding result on the existence of a rearrangement of $\Phi $ is shown, too. The proofs are based on moment conditions (e.g. the concept of weak multiplicativity) instead of martingale techniques, which are used e.g. by D. J. Aldous or S. D. Chatterji.

AMS Subject Classification (1991): 60F05

Keyword(s): Central limit theorem, subsequences, moment conditions

Received December 6, 1991. (Registered under 5541/2009.)