## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 1 (1974), 49-62.

### The Invariance Principle for One-Sample Rank-Order Statistics

#### Abstract

Analogous to the Donsker theorem on partial cumulative sums of independent random variables, for a broad class of one-sample rank order statistics, weak convergence to Brownian motion processes is studied here. A simple proof of the asymptotic normality of these statistics for random sample sizes is also presented. Some asymptotic results on renewal theory for one-sample rank order statistics are derived.

#### Article information

**Source**

Ann. Statist., Volume 2, Number 1 (1974), 49-62.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342612

**Digital Object Identifier**

doi:10.1214/aos/1176342612

**Mathematical Reviews number (MathSciNet)**

MR345309

**Zentralblatt MATH identifier**

0273.60005

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B10: Convergence of probability measures

Secondary: 62G99: None of the above, but in this section

**Keywords**

Brownian motion processes invariance principles martingales one-sample rank order statistics random sample sizes renewal theory weak convergence

#### Citation

Sen, Pranab Kumar. The Invariance Principle for One-Sample Rank-Order Statistics. Ann. Statist. 2 (1974), no. 1, 49--62. doi:10.1214/aos/1176342612. https://projecteuclid.org/euclid.aos/1176342612

#### Corrections

- See Correction: Pranab Kumar Sen. Notes: Correction to "The Invariance Principle for One-Sample Rank-Order Statistics". Ann. Statist., Volume 2, Number 6 (1974), 1358--1358.Project Euclid: euclid.aos/1176342893