## Bernoulli

- Bernoulli
- Volume 17, Number 2 (2011), 671-686.

### Limit theorems for functions of marginal quantiles

G. Jogesh Babu, Zhidong Bai, Kwok Pui Choi, and Vasudevan Mangalam

#### Abstract

Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur’s representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that $$\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^{n}\phi\bigl(X_{n:i}^{(1)},\ldots,X_{n:i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}Z_{n,i}+\mathrm{o}_{P}(1)$$ as $n → ∞$, where $γ̄>$ is a constant and $Z_{n,i}$ are i.i.d. random variables for each $n$. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.

#### Article information

**Source**

Bernoulli, Volume 17, Number 2 (2011), 671-686.

**Dates**

First available in Project Euclid: 5 April 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1302009242

**Digital Object Identifier**

doi:10.3150/10-BEJ287

**Mathematical Reviews number (MathSciNet)**

MR2787610

**Zentralblatt MATH identifier**

1253.60024

**Keywords**

central limit theorem Cramér–Wold device lost association quantiles strong law of large numbers weak convergence of a process

#### Citation

Babu, G. Jogesh; Bai, Zhidong; Choi, Kwok Pui; Mangalam, Vasudevan. Limit theorems for functions of marginal quantiles. Bernoulli 17 (2011), no. 2, 671--686. doi:10.3150/10-BEJ287. https://projecteuclid.org/euclid.bj/1302009242