Donsker's theorem (the invariance principle)

The ordinary central limit theorem says: add up many small independent shocks and the SUM looks Normal. Donsker's theorem upgrades this from a single endpoint to the whole PATH. Take a random walk — a particle that takes a +1 or -1 step at each tick — and instead of asking only where it ends up, watch the entire jagged trajectory. Donsker's theorem says that if you shrink the steps and speed up time the right way, that staircase trajectory, viewed as a curve, converges to a Brownian motion. It is the reason Brownian motion is the universal continuous-time limit: it is the CLT for whole journeys, not just destinations.

Here is the right way to shrink and speed up. Let S_n be the position of a mean-zero, variance-one random walk after n steps. Build a continuous curve W_n(t) by plotting S_k at time k/n and joining the dots, then squash the vertical scale by dividing by sqrt(n): W_n(t) = S_(floor(nt)) / sqrt(n). The square-root squashing is exactly the CLT scaling — it keeps the variance at each time t equal to t. As n grows, this rescaled walk W_n converges (in distribution, on the space of continuous paths) to a standard Brownian motion B. The classical CLT is just the snapshot of this at the single time t = 1.

The name 'invariance principle' captures the punchline: the limit does NOT depend on the fine details of the step distribution. Whether your steps are plus-or-minus one, Normal, or any mean-zero finite-variance shape, the rescaled path converges to the SAME Brownian motion. This universality is why one object — Brownian motion — models so many different microscopic randomnesses, and it lets you compute limiting laws for complicated walk functionals (like the maximum, or the time spent positive) by computing them for Brownian motion instead. The catch is the same as for the CLT: finite variance is required; heavy-tailed steps converge to different, jumpy limits instead.