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

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