Limit theory for the empirical extremogram of random fields

TL;DR

This paper develops limit theory for the empirical extremogram, a tool to measure extremal dependence in random fields, providing conditions for its asymptotic normality across various models.

Contribution

It establishes asymptotic normality results for the empirical extremogram under broad conditions, including max-stable processes with Fréchet margins.

Findings

Asymptotic normality of the empirical extremogram is proven under specific conditions.

Results apply to spatial, space-time processes, and time series models.

Applications include max-moving average and Brown-Resnick processes.

Abstract

Regularly varying stochastic processes are able to model extremal dependence between process values at locations in random fields. We investigate the empirical extremogram as an estimator of dependence in the extremes. We provide conditions to ensure asymptotic normality of the empirical extremogram centred by a pre-asymptotic version. For max-stable processes with Fr{\'e}chet margins we provide conditions such that the empirical extremogram centred by its true version is asymptotically normal. The results of this paper apply to a variety of spatial and space-time processes, and to time series models. We apply our results to max-moving average processes and Brown-Resnick processes.