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.
Let
The central limit theorem asserts that
History
Let Fn be the empirical distribution function of the sequence of i.i.d. random variables
indexed by x ∈ R. By the classical central limit theorem, for fixed x, the random variable Gn(x) converges in distribution to a Gaussian (normal) random variable G(x) with zero mean and variance F(x)(1 − F(x)) as the sample size n grows.
Theorem (Donsker, Skorokhod, Kolmogorov) The sequence of Gn(x), as random elements of the Skorokhod space
The process G(x) can be written as B(F(x)) where B is a standard Brownian bridge on the unit interval.
Kolmogorov (1933) showed that when F is continuous, the supremum
In 1952 Donsker stated and proved (not quite correctly) 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].
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 cadlag 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
Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove that there exist Xi, iid uniform in [0,1] and a sequence of sample-continuous Brownian bridges Bn, such that
is measurable and converges in probability to 0. An improved version of this result, providing more detail on the rate of convergence, is the Komlós–Major–Tusnády approximation.