## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 1 (1973), 146-152.

### Convergence of Reduced Empirical and Quantile Processes with Application to Functions of Order Statistics in the Non-I.I.D. Case

#### Abstract

Any triangular array of row independent $\mathrm{rv}$'s having continuous $\mathrm{df}$'s can be transformed naturally so that the empirical and quantile processes of the resulting $\mathrm{rv}$'s are relatively compact. Moreover, convergence (to a necessarily normal process) takes place if and only if a simple covariance function converges pointwise. Using these results we derive the asymptotic normality of linear combinations of functions of order statistics of non-i.i.d. $\mathrm{rv}$'s in the case of bounded scores.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 1 (1973), 146-152.

**Dates**

First available in Project Euclid: 25 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1193342391

**Digital Object Identifier**

doi:10.1214/aos/1193342391

**Mathematical Reviews number (MathSciNet)**

MR336776

**Zentralblatt MATH identifier**

0255.62044

**Subjects**

Primary: 62G30: Order statistics; empirical distribution functions

Secondary: 60B10: Convergence of probability measures 62E20: Asymptotic distribution theory

**Keywords**

Convergence of non-i.i.d. empirical processes asymptotic normality of functions of non-i.i.d. order statistics

#### Citation

Shorack, Galen R. Convergence of Reduced Empirical and Quantile Processes with Application to Functions of Order Statistics in the Non-I.I.D. Case. Ann. Statist. 1 (1973), no. 1, 146--152. doi:10.1214/aos/1193342391. https://projecteuclid.org/euclid.aos/1193342391