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.
Theory.
Let formula_1 be independent uniform (0,1) random variables. Define a uniform empirical distribution function as
formula_2
Define a uniform empirical process as
formula_3
The Donsker theorem (1952) shows that formula_4 converges in law to a Brownian bridge formula_5 Komlós, Major and Tusnády established a sharp bound for the speed of this weak convergence.
Theorem (KMT, 1975) On a suitable probability space for independent uniform (0,1) r.v. formula_6 the empirical process formula_7 can be approximated by a sequence of Brownian bridges formula_8 such that
formula_9
for all positive integers "n" and all formula_10, where "a", "b", and "c" are positive constants.
Corollary.
A corollary of that theorem is that for any real iid r.v. formula_11 with cdf formula_12 it is possible to construct a probability space where independent sequences of empirical processes formula_13 and Gaussian processes formula_14 exist such that
 formula_15     almost surely.