Komlós–Major–Tusnády approximation

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

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Let   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.

Theorem (KMT, 1975) On a suitable probability space for independent uniform (0,1) r.v.   the empirical process   can be approximated by a sequence of Brownian bridges   such that
 
for all positive integers n and all  , where a, b, and c are positive constants.

Corollary

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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[clarification needed] sequences of empirical processes   and Gaussian processes   exist such that

      almost surely.

References

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  • Komlos, J., Major, P. and Tusnady, G. (1975) An approximation of partial sums of independent rv’s and the sample df. I, Wahrsch verw Gebiete/Probability Theory and Related Fields, 32, 111–131. doi:10.1007/BF00533093
  • Komlos, J., Major, P. and Tusnady, G. (1976) An approximation of partial sums of independent rv’s and the sample df. II, Wahrsch verw Gebiete/Probability Theory and Related Fields, 34, 33–58. doi:10.1007/BF00532688