In theory of probability, the Komlós–Major–Tusnády approximation (also known as the KMT approximation, the KMT embedding, or the Hungarian embedding) 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.
Let U 1 , U 2 , … be independent uniform (0,1) random variables. Define a uniform empirical distribution function as
F U , n ( t ) = 1 n ∑ i = 1 n 1 U i ≤ t , t ∈ [ 0 , 1 ] . Define a uniform empirical process as
α U , n ( t ) = n ( F U , n ( t ) − t ) , t ∈ [ 0 , 1 ] . The Donsker theorem (1952) shows that α U , n ( t ) converges in law to a Brownian bridge B ( t ) . 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.
U 1 , U 2 … the empirical process
{ α U , n ( t ) , 0 ≤ t ≤ 1 } can be approximated by a sequence of Brownian bridges
{ B n ( t ) , 0 ≤ t ≤ 1 } such that
P { sup 0 ≤ t ≤ 1 | α U , n ( t ) − B n ( t ) | > 1 n ( a log n + x ) } ≤ b e − c x for all positive integers
n and all
x > 0 , where
a,
b, and
c are positive constants.
A corollary of that theorem is that for any real iid r.v. X 1 , X 2 , … , with cdf F ( t ) , it is possible to construct a probability space where independent sequences of empirical processes α X , n ( t ) = n ( F X , n ( t ) − F ( t ) ) and Gaussian processes G F , n ( t ) = B n ( F ( t ) ) exist such that
lim sup n → ∞ n ln n ∥ α X , n − G F , n ∥ ∞ < ∞ , almost surely.
