Donsker's theorem
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 be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let . The stochastic process is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by The central limit theorem asserts that converges in distribution to a standard Gaussian random variable as . Donsker's invariance principle extends this convergence to the whole function . More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space , the random function converges in distribution to a standard Brownian motion as Formal statement 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 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