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Proof of the central limit theorem

For a théorem of such fundamental importance to statistics and applied probability, the central limit théorem has a remarkably simple proof using characteristic functions. It is similar to the proof of a (wéak) law of large numbers. For any variabel acak, Y, with zero méan and unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theorem,

Letting Yi be (Xi − μ)/σ, the standardised value of Xi, it is éasy to see that the standardised méan of the observations X1, X2, ..., Xn is just

By simple properties of characteristic functions, the characteristic function of Zn is

But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit théorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.