Komlós–Major–Tusnády approximation

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 $U_{1},U_{2},\ldots$ be independent uniform (0,1) random variables. Define a uniform empirical distribution function as

F_{{U,n}}(t)={\frac {1}{n}}\sum _{{i=1}}^{n}{\mathbf {1}}_{{U_{i}\leq t}},\quad t\in [0,1].

Define a uniform empirical process as

\alpha _{{U,n}}(t)={\sqrt {n}}(F_{{U,n}}(t)-t),\quad t\in [0,1].

The Donsker theorem (1952) shows that $\alpha _{{U,n}}(t)$ 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}\ldots

the empirical process

\{\alpha _{{U,n}}(t),0\leq t\leq 1\}

can be approximated by a sequence of Brownian bridges

\{B_{n}(t),0\leq t\leq 1\}

such that

P\left\{\sup _{{0\leq t\leq 1}}|\alpha _{{U,n}}(t)-B_{n}(t)|>{\frac {1}{{\sqrt {n}}}}(a\log n+x)\right\}\leq be^{{-cx}}

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. $X_{1},X_{2},\ldots ,$ with cdf $F(t),$ it is possible to construct a probability space where independent sequences of empirical processes $\alpha _{{X,n}}(t)={\sqrt {n}}(F_{{X,n}}(t)-F(t))$ and Gaussian processes $G_{{F,n}}(t)=B_{n}(F(t))$ exist such that

\limsup _{{n\to \infty }}{\frac {{\sqrt {n}}}{\ln n}}{\big \|}\alpha _{{X,n}}-G_{{F,n}}{\big \|}_{\infty }<\infty ,

almost surely.

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. 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

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