Donsker's Invariance Principle

The Central Limit Theorem for the whole path: every mean-zero walk rescales to the same Brownian motion.

The idea

Theorem (Donsker's invariance principle).

Let $S_{n} = X_{1} + \cdots + X_{n}$ be a random walk whose steps are independent and identically distributed with mean $0$ and finite variance $\sigma^{2}$. Rescale the walk into a path on the unit time interval, $X_{n}(t) = \frac{S_{\lfloor nt \rfloor}}{\sigma\sqrt{n}}, \qquad 0 \le t \le 1,$ joining consecutive values by straight lines so that $X_{n}$ is a continuous function. Then $X_{n}$ converges in distribution to standard Brownian motion on $[0, 1]$.

The Central Limit Theorem already governs the endpoint: the single value $X_{n}(1)$ becomes normal. But most questions about a walk — how high it climbed, when it first crossed a level, how long it stayed positive — depend on the whole path, and the endpoint does not determine them. Donsker's principle converges the whole random function at once. We can therefore answer a path question for Brownian motion, where calculus is available, and the answer holds for every walk satisfying the hypotheses.

That transfer is the invariance in the name. The steps may be coin flips, uniform draws, or heavily skewed; only their mean and variance survive into the limit, and the limit is the same Brownian motion in every case.

Ways to work on it

• Walkthrough. How rescaled random walks converge to Brownian motion, and what convergence of whole paths adds to the Central Limit Theorem.
• Practice. Rescale a random walk correctly, check the theorem's hypotheses, and compute a limiting probability.
• Hardest. Find the limiting behavior of a random walk's maximum — including for a walk with drift.

Not sure where to start? Take the ten-question placement test.