---
tag: 818832PMSV
kind: statement
role: theorem
name: Donsker's theorem
msc: [60F17]
uses: [00MRN0FARV, 0SRYJGXKR4, 22735FGJV6, 0X3WGX95FE, 04VZE3FZXF]
about: [22735FGJV6, 04VZE3FZXF]
sources:
  - url: https://en.wikipedia.org/wiki/Donsker%27s_theorem
same_as:
  wikidata: Q3527064
  wikipedia: Donsker's theorem
---

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables with $E[X_1] = 0$ and $\sigma^2 = E[X_1^2] \in (0, \infty)$.
Let $S_0 = 0$ and $S_k = X_1 + \cdots + X_k$.
For $n \ge 1$ and $t \in [0, 1]$, let
$$W_n(t) = \frac{1}{\sigma \sqrt{n}} \left( S_{\floor{nt}} + (nt - \floor{nt}) \, X_{\floor{nt} + 1} \right),$$
the linear interpolation of the rescaled [[random walk|04VZE3FZXF]].
Let $C[0, 1]$ be the space of [[continuous|0X3WGX95FE]] functions on $[0, 1]$ with the supremum norm.
Then $W_n$ [[converges in distribution|0SRYJGXKR4]], as random elements of $C[0, 1]$, to a standard [[Wiener process|22735FGJV6]] $(W(t))_{t \in [0, 1]}$.

## Remarks

The proof has two steps.
The [[central limit theorem|0EPZC4Z1FC]], 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|MBW9JCGZC1]].

Example: the supremum map $f \mapsto \max_{t \in [0, 1]} f(t)$ is continuous on $C[0, 1]$.
By the [[continuous mapping theorem|CFRN0G7GDB]], $\max_{k \le n} S_k / (\sigma \sqrt{n})$ converges in distribution to $\max_{t \le 1} W(t)$.
By the [[reflection principle|K6R7GHB3VS]], 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|XKFG1676CG]].
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|3MCVMBAGD1]], in a suitable space of functions with jumps.

## Explanations

### L1 human

Draw the path of a random walk with many small steps and shrink it so that it fits in a unit box.
As the steps get smaller, the picture looks more and more like the path of Brownian motion.
This holds for any step law with mean zero and finite variance, so Brownian motion is a universal limit.

### L2 human

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 $F$.