In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem for empirical distribution functions. Specifically, the theorem states that an appropriately centered and scaled version of the empirical distribution function converges to a Gaussian process.
Let Fn be the empirical distribution function of the sequence of i.i.d. random variables with distribution function F. Define the centered and scaled version of Fn by
In 1952 Donsker stated and proved (not quite correctly)[4] a general extension for the Doob–Kolmogorov heuristic approach. In the original paper, Donsker proved that the convergence in law of Gn to the Brownian bridge holds for Uniform[0,1] distributions with respect to uniform convergence in t over the interval [0,1].[2]
However Donsker's formulation was not quite correct because of the problem of measurability of the functionals of discontinuous processes. In 1956 Skorokhod and Kolmogorov defined a separable metric d, called the Skorokhod metric, on the space of càdlàg functions on [0,1], such that convergence for d to a continuous function is equivalent to convergence for the sup norm, and showed that Gn converges in law in to the Brownian bridge.
Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove[4] that there exist Xi, iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges Bn, such that
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