Komlós–Major–Tusnády approximation

In probability theory, the Komlós–Major–Tusnády approximation approximation of random walk by a standard Brownian motion constructed on the same probability space, and 2) an approximation of the empirical process by a Brownian bridge constructed on the same probability space. It is named after Hungarian mathematicians János Komlós, Gábor Tusnády, and Péter Major, who proved it in 1975.

Theory

Let be independent uniform random variables. Define a uniform empirical distribution function as
Define a uniform empirical process as
The Donsker theorem shows that converges in law to a Brownian bridge Komlós, Major and Tusnády established a sharp bound for the speed of this weak convergence.

Corollary

A corollary of that theorem is that for any real iid r.v. with cdf it is possible to construct a probability space where independent sequences of empirical processes and Gaussian processes exist such that