Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

An almost everywhere central limit theorem

Gunnar A. Brosamlera1a2

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, S0305004100065750_xs1D504, P) with S0305004100065750_inline001 we have weak convergence of the distributions of S0305004100065750_inline002 to the standard normal distribution on xs211D. 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

S0305004100065750_eqnU001

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)