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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 [1, 2].

Let X1, X2, X3, ... 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

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 D[0, 1], the random function