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) 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, , and , who proved it in 1975.

Theory[]

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

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

Define a uniform empirical process as

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

The Donsker theorem (1952) shows that ${\displaystyle \alpha _{U,n}(t)}$ converges in law to a Brownian bridge ${\displaystyle B(t).}$ 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. ${\displaystyle U_{1},U_{2}\ldots }$ the empirical process ${\displaystyle \{\alpha _{U,n}(t),0\leq t\leq 1\}}$ can be approximated by a sequence of Brownian bridges ${\displaystyle \{B_{n}(t),0\leq t\leq 1\}}$ such that

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

for all positive integers n and all ${\displaystyle x>0}$, where a, b, and c are positive constants.

Corollary[]

A corollary of that theorem is that for any real iid r.v. ${\displaystyle X_{1},X_{2},\ldots ,}$ with cdf ${\displaystyle F(t),}$ it is possible to construct a probability space where independent^{[clarification needed]} sequences of empirical processes ${\displaystyle \alpha _{X,n}(t)={\sqrt {n}}(F_{X,n}(t)-F(t))}$ and Gaussian processes ${\displaystyle G_{F,n}(t)=B_{n}(F(t))}$ exist such that

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