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 X 1 , X 2 , X 3 , … {\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 S n := ∑ i = 1 n X i {\displaystyle S_{n}:=\sum _{i=1}^{n}X_{i}} . The stochastic process S := ( S n ) n ∈ N {\displaystyle S:=(S_{n})_{n\in \mathbb {N} }} is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by W ( n ) ( t ) := S ⌊ n t ⌋ n , t ∈ [ 0 , 1 ] . {\displaystyle W^{(n)}(t):={\frac {S_{\lfloor nt\rfloor }}{\sqrt {n}}},\qquad t\in [0,1].} The central limit theorem asserts that W ( n ) ( 1 ) {\displaystyle W^{(n)}(1)} converges in distribution to a standard Gaussian random variable W ( 1 ) {\displaystyle W(1)} as n → ∞ {\displaystyle n\to \infty } . Donsker's invariance principle extends this convergence to the whole function W ( n ) := ( W ( n ) ( t ) ) t ∈ [ 0 , 1 ] {\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 D [ 0 , 1 ] {\displaystyle {\mathcal {D}}[0,1]} , the random function W ( n ) {\displaystyle W^{(n)}} converges in distribution to a standard Brownian motion W := ( W ( t ) ) t ∈ [ 0 , 1...