In probability theory, the Komlós–Major–Tusnády approximation (also known as the KMT approximation, the KMT embedding, or the Hungarian embedding) refers to one of the two strong embedding theorems: 1) 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.
be independent uniform (0,1) random variables.
Define a uniform empirical distribution function as Define a uniform empirical process as The Donsker theorem (1952) shows that
converges in law to a Brownian bridge
Komlós, Major and Tusnády established a sharp bound for the speed of this weak convergence.
A corollary of that theorem is that for any real iid r.v.
it is possible to construct a probability space where independent[clarification needed] sequences of empirical processes
and Gaussian processes