Donsker's theorem

Remarks

The proof has two steps. The central limit theorem, applied to the independent increments, gives the convergence of the finite-dimensional distributions of $W_n$ to those of $W$. A maximal inequality then shows that the laws of the $W_n$ are tight in $C[0, 1]$, and tightness together with convergence of finite-dimensional distributions gives weak convergence.

Example: the supremum map $f \mapsto \max_{t \in [0, 1]} f(t)$ is continuous on $C[0, 1]$. By the continuous mapping theorem, $\max_{k \le n} S_k / (\sigma \sqrt{n})$ converges in distribution to $\max_{t \le 1} W(t)$. By the reflection principle, this limit has the law of $\abs{W(1)}$. So the limit does not depend on the law of $X_1$, which is why the result is called an invariance principle.

A second form of the theorem concerns the empirical process. If $U_1, U_2, \dots$ are i.i.d. uniform on $[0, 1]$ with empirical distribution function $F_n$, then $\sqrt{n}(F_n(t) - t)$ converges in distribution to a Brownian bridge, in a suitable space of functions with jumps.

Going deeper

For a student

Donsker's theorem is the functional form of the central limit theorem. With the continuous mapping theorem, it gives the limit laws of functionals of random walks, computed once for Brownian motion. Examples are the maximum, the time spent above zero and the time of the last zero. In statistics, the empirical-process form gives the limit law of the Kolmogorov–Smirnov statistic $\sqrt{n} \sup_x \abs{F_n(x) - F(x)}$ for a continuous $F$.