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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

Let U1,U2, be independent uniform (0,1) random variables. Define a uniform empirical distribution function as

FU,n(t)=1ni=1n1Uit,t[0,1].

Define a uniform empirical process as

αU,n(t)=n(FU,n(t)t),t[0,1].

The Donsker theorem (1952) shows that αU,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. U1,U2 the empirical process {αU,n(t),0t1} can be approximated by a sequence of Brownian bridges {Bn(t),0t0} such that
P{sup0t1|αU,n(t)Bn(t)|>1n(alogn+x)}becx
for all positive integers n and all x>0, where a, b, and c are positive constants.

Corollary

A corollary of that theorem is that for any real iid r.v. X1,X2,, with cdf F(t), it is possible to construct a probability space where independentTemplate:Clarify sequences of empirical processes αX,n(t)=n(FX,n(t)F(t)) and Gaussian processes GF,n(t)=Bn(F(t)) exist such that

lim supnnlnnαX,nGF,n<,     almost surely.

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References

  • 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. Electronic Instrument Positions Staff (Standard ) Cameron from Clarence Creek, usually spends time with hobbies and interests which include knotting, property developers in singapore apartment For sale and boomerangs. Has enrolled in a world contiki journey. Is extremely thrilled specifically about visiting .
  • 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. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.