## Afrika Statistika

- Afr. Stat.
- Volume 13, Number 3 (2018), 1795-1822.

### A Central limit Theorem of dependent sums of standard exponential functionals motivated by extreme value theory

Gane Samb Lo, Adja Mbarka Fall, and Harouna Sangaré

#### Abstract

Consider the following demimartingale \begin{equation*} \sum_{j=1}^{k-1}f(j)(\exp (-\gamma \sum_{h=j+1}^{k-1}E_{h}/h)-\exp (-\gamma \sum_{h=j}^{k-1}E_{h}/h)), \end{equation*} where \(E_{1},E_{2},...\) are independent standard exponential random variables, \(\gamma>0\), \(k\) is a positive integer and \(f(j)\) is an increasing function of the integer $j\geq1$. We find general conditions under which the central limit theorem (CLT) holds and next apply the results to find the asymptotic behavior of the functional Hill within the Extreme Value Theory (EVT) field. This results show a new trend for the central limit theorem issue for non-stationary sequences of associated variables.

#### Article information

**Source**

Afr. Stat., Volume 13, Number 3 (2018), 1795-1822.

**Dates**

First available in Project Euclid: 12 December 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.as/1544583716

**Digital Object Identifier**

doi:10.16929/as/1795.134

**Mathematical Reviews number (MathSciNet)**

MR3887184

**Zentralblatt MATH identifier**

07003217

**Subjects**

Primary: 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators 60F05: Central limit and other weak theorems

Secondary: 60B10: Convergence of probability measures 60F17: Functional limit theorems; invariance principles

**Keywords**

Extreme value theory Associated random variables demimartingales asymptotic laws functional Hill processes extreme value theory statistical tests

#### Citation

Lo, Gane Samb; Fall, Adja Mbarka; Sangaré, Harouna. A Central limit Theorem of dependent sums of standard exponential functionals motivated by extreme value theory. Afr. Stat. 13 (2018), no. 3, 1795--1822. doi:10.16929/as/1795.134. https://projecteuclid.org/euclid.as/1544583716