The Annals of Probability

Weighted Empirical and Quantile Processes

Miklos Csorgo, Sandor Csorgo, Lajos Horvath, and David M. Mason

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Abstract

We introduce a new Brownian bridge approximation to weighted empirical and quantile processes with rates in probability. This approximation leads to a number of general invariance theorems for empirical and quantile processes indexed by functions. Improved versions of the Chibisov-O'Reilly theorems, the Eicker-Jaeschke theorems for standardized empirical and quantile processes, the normal convergence criterion, and various other old and new asymptotic results on empirical and quantile processes are presented as consequences of our general theorems. In the process, we provide a new characterization of Erdos-Feller-Kolmogorov-Petrovski upper-class functions for the Brownian motion in an improved form.

Article information

Source
Ann. Probab., Volume 14, Number 1 (1986), 31-85.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992617

Digital Object Identifier
doi:10.1214/aop/1176992617

Mathematical Reviews number (MathSciNet)
MR815960

Zentralblatt MATH identifier
0589.60029

JSTOR
links.jstor.org

Subjects
Primary: 60F99: None of the above, but in this section
Secondary: 60F17: Functional limit theorems; invariance principles 60J65: Brownian motion [See also 58J65] 60F05: Central limit and other weak theorems 60F20: Zero-one laws 62G30: Order statistics; empirical distribution functions

Keywords
Weighted empirical and quantile processes Brownian bridge approximations weak invariance principles indexed by functions

Citation

Csorgo, Miklos; Csorgo, Sandor; Horvath, Lajos; Mason, David M. Weighted Empirical and Quantile Processes. Ann. Probab. 14 (1986), no. 1, 31--85. doi:10.1214/aop/1176992617. https://projecteuclid.org/euclid.aop/1176992617


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