Komlós–Major–Tusnády approximation

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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.

Theory[edit]

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[edit]

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[edit]

  • 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