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 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 ${\displaystyle X_{1},X_{2},X_{3},\ldots }$ 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 (partial-sum process) by

${\displaystyle W^{(n)}(t):={\frac {S_{\lfloor nt\rfloor }}{\sqrt {n}}},\qquad t\in [0,1].}$

The central limit theorem asserts that ${\displaystyle W^{(n)}(1)}$ converges in distribution to a standard Gaussian random variable ${\displaystyle W(1)}$ as ${\displaystyle n\to \infty }$. Donsker's invariance principle extends this convergence to the whole function ${\displaystyle W^{(n)}:=(W^{(n)}(t))_{t\in [0,1]}}$. More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space ${\displaystyle {\mathcal {D}}[0,1]}$, the random function ${\displaystyle W^{(n)}}$ converges in distribution to a standard Brownian motion ${\displaystyle W:=(W(t))_{t\in [0,1]}}$ as ${\displaystyle n\to \infty .}$