Komlós–Major–Tusnády approximation
In theory of probability, the Komlós–Major–Tusnády approximation is an approximation of the empirical process by a Gaussian process constructed on the same probability space. It is named after Hungarian mathematicians János Komlós, Gábor Tusnády, and Péter Major.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
