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Komlós–Major–Tusnády approximation

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

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

Define a uniform empirical processas

${\displaystyle \alpha _{U,n}($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[edit]

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[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