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 ${\displaystyle X_{1},X_{2},X_{3},\ldots }$ be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let ${\displaystyle S_{n}:=\sum _{i=1}^{n}X_{i}}$. The stochastic process ${\displaystyle S:=(S_{n})_{n\in \mathbb {N} }}$ is known as a random walk. Define the diffusively rescaled random walk by

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