Palm theory for extremes of stationary regularly varying time series and random fields

TL;DR

This paper develops a Palm theory-inspired framework for understanding the extremal behavior of stationary regularly varying random fields, characterizing tail processes and their relation to typical extreme clusters.

Contribution

It introduces invariance properties and dualities for tail processes, providing a new theoretical foundation for analyzing extremal clusters in random fields.

Findings

Characterization of tail processes via exceedance-stationarity and polar decomposition.

Duality relations between tail process and anchored tail process.

Distributional results for typical extremal clusters in moving average models.

Abstract

The tail process $\boldsymbol{Y}=(Y_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$ of a stationary regularly varying random field $\boldsymbol{X}=(X_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$ represents the asymptotic local distribution of $\boldsymbol{X}$ as seen from its typical exceedance over a threshold $u$ as $u\to\infty$. Motivated by the standard Palm theory, we show that every tail process satisfies an invariance property called exceedance-stationarity and that this property, together with the polar decomposition of the tail process, characterizes the class of all tail processes. We then restrict to the case when $Y_{\boldsymbol{i}}\to 0$ as $|\boldsymbol{i}|\to\infty$ and establish a couple of Palm-like dualities between the tail process and the so-called anchored tail process which, under suitable conditions, represents the asymptotic distribution of a typical cluster of extremes of $\boldsymbol{X}$. The main message is that the distribution of the tail process is biased towards clusters with more exceedances. Finally, we use these results to determine the distribution of the typical cluster of extremes for moving average processes with random coefficients and heavy-tailed innovations.