DonskersTheorem

Donsker's theorem, also known as the invariance principle or Donsker's invariance principle, is a fundamental result in probability theory and is named after Monroe D. Donsker. It provides a functional central limit theorem for stochastic processes and is a powerful tool in the study of weak convergence.

Consider a sequence of independent and identically distributed (i.i.d.) random variables $X_1, X_2, \ldots$, each with mean 0 and variance 1. Let

$$ S_n = \frac{X_1 + X_2 + \ldots + X_n}{\sqrt{n}}. $$

Donsker's theorem states that, when appropriately normalized, the cumulative sum process converges in distribution to a Brownian motion.

To make this precise, define a stochastic process

$$W_n(t) = \frac{1}{\sqrt{n}} \sum_{k=1}^{\lfloor nt \rfloor} X_k \forall 0 \leq t \leq 1$$

Then, under certain conditions on the $X_i$, Donsker's theorem says that as $n \to \infty$, the process ${ W_n(t) }_{0 \leq t \leq 1}$ converges in distribution to a standard Brownian motion on the interval $[0, 1]$ in the Skorokhod space (space of càdlàg functions).

Donsker's theorem has several applications in probability theory, statistics, and stochastic processes. It plays a central role in the theory of empirical processes and has been used in statistical mechanics, mathematical finance, and various areas of applied mathematics.