## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 1 (1989), 239-256.

### Hungarian Constructions from the Nonasymptotic Viewpoint

#### Abstract

Let $x_1, \ldots, x_n$ be independent random variables with uniform distribution over $\lbrack 0, 1\rbrack$, defined on a rich enough probability space $\Omega$. Denoting by $\hat{\mathbb{F}}_n$ the empirical distribution function associated with these observations and by $\alpha_n$ the empirical Brownian bridge $\alpha_n(t) = \sqrt n(\hat{\mathbb{F}}_n(t) - t)$, Komlos, Major and Tusnady (KMT) showed in 1975 that a Brownian bridge $\mathbb{B}^0$ (depending on $n$) may be constructed on $\Omega$ in such a way that the uniform deviation $\|\alpha_n - \mathbb{B}^0\|_\infty$ between $\alpha_n$ and $\mathbb{B}^0$ is of order of $\log(n)/\sqrt n$ in probability. In this paper, we prove that a Poisson bridge $\mathbb{L}^0_n$ may be constructed on $\Omega$ (note that this construction is not the usual one) in such a way that the uniform deviations between any two of the three processes $\alpha_n, \mathbb{L}^0_n$ and $\mathbb{B}^0$ are of order of $\log(n)/\sqrt n$ in probability. Moreover, we give explicit exponential bounds for the error terms, intended for asymptotic as well as nonasymptotic use.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 1 (1989), 239-256.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991506

**Digital Object Identifier**

doi:10.1214/aop/1176991506

**Mathematical Reviews number (MathSciNet)**

MR972783

**Zentralblatt MATH identifier**

0667.60042

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60F99: None of the above, but in this section 62G30: Order statistics; empirical distribution functions

**Keywords**

Brownian bridge empirical process Poisson bridge strong approximation

#### Citation

Bretagnolle, J.; Massart, P. Hungarian Constructions from the Nonasymptotic Viewpoint. Ann. Probab. 17 (1989), no. 1, 239--256. doi:10.1214/aop/1176991506. https://projecteuclid.org/euclid.aop/1176991506