1 Answers
In probability theory, Donsker's theorem , named after Monroe D. Donsker, is a functional extension of the central limit theorem.
Let X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\ldots } be a sequence of independent and identically distributed 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 := n ∈ N {\displaystyle S:=_{n\in \mathbb {N} }} is known as a random walk. Define the diffusively rescaled random walk by
The central limit theorem asserts that W {\displaystyle W^{}} converges in distribution to a standard Gaussian random variable W {\displaystyle W} as n → ∞ {\displaystyle n\to \infty }. Donsker's invariance principle extends this convergence to the whole function W := ] t ∈ {\displaystyle W^{}:=}]_{t\in }}. More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space D {\displaystyle {\mathcal {D}}} , the random function W {\displaystyle W^{}} converges in distribution to a standard Brownian motion W := ] t ∈ {\displaystyle W:=]_{t\in }} as n → ∞ . {\displaystyle n\to \infty.}