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

Abstract

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

k-1              k-1                  k-1

∑f(j)(exp(-y ∑E_h/h) -exp(-y ∑E_h/h)),

j=1            h=j+1               h+1

where E₁,E₂, ... are independent standard exponential random variables, y > 0, k is a positive integer and f(j) is an increasing function of the integer j ≥ 1. 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.

Keywords: Extreme value theory; Associated random variables; demimartingales; asymptotic laws; functional Hill processes; extreme value theory; statistical tests.

AMS 2010 Mathematics Subject Classification Objects: Primary 62E20; 62F12; 60F05. Secondary 60B10; 60F17