The integrated periodogram of a dependent extremal event sequence

TL;DR

This paper develops a functional central limit theorem for the integrated periodogram of dependent extremal events, enabling goodness-of-fit testing for extreme value models using a bootstrap approach.

Contribution

It introduces a new asymptotic theory for the integrated periodogram of extremal events in dependent sequences, including a bootstrap method for distribution approximation.

Findings

The limiting process is a Gaussian process with a complex covariance structure.

A stationary bootstrap procedure effectively approximates the distribution of the limiting process.

The methodology applies to real and simulated data for goodness-of-fit testing.

Abstract

We investigate the asymptotic properties of the integrated periodogram calculated from a sequence of indicator functions of dependent extremal events. An event in Euclidean space is extreme if it occurs far away from the origin. We use a regular variation condition on the underlying stationary sequence to make these notions precise. Our main result is a functional central limit theorem for the integrated periodogram of the indicator functions of dependent extremal events. The limiting process is a continuous Gaussian process whose covari- ance structure is in general unfamiliar, but in the iid case a Brownian bridge appears. In the general case, we propose a stationary bootstrap procedure for approximating the distribution of the limiting process. The developed theory can be used to construct classical goodness-of-fit tests such as the Grenander- Rosenblatt and Cram\'{e}r-von Mises tests which are based only on the extremes in the sample. We apply the test statistics to simulated and real-life data.