Abstract
We prove the central limit theorem from the definition of weak convergence using the Haar basis, calculus, and elementary probability, and we estimate the rate of convergence off the tails. The use of the Haar basis pinpoints the role of $L^{2}([0,1])$ in the CLT as well as the assumption of finite variance.
Citation
Vladimir Dobrić. Patricia Garmirian. "A new direct proof of the central limit theorem." Illinois J. Math. 61 (3-4) 355 - 370, Fall and Winter 2017. https://doi.org/10.1215/ijm/1534924830
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