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An almost everywhere central limit theorem

  • Gunnar A. Brosamler (a1) (a2)

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 ℝ. 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.

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[2] G. A. Brosamler . The asymptotic behaviour of certain additive functionals of Brownian motion. Invent. Math. 20 (1973), 8796.

[7] A. Fisher . Convex-invariant means and a pathwise central limit theorem. Adv. in Math. 63 (1987), 213246.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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