a1 Fachbereich Mathematik, Universität des Saarlandes, Saarbrücken, West Germany
a2 Department of Mathematics, The University of British Columbia, Vancouver, B.C., Canada
The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on . We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean that
for all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.
(Received November 24 1987)
(Revised January 28 1988)