Spectral Measure and Tail Process for Multivariate Dynamical Extremes

Abstract

We study the asymptotic behavior of extreme events in multivariate stationary sequences, with particular emphasis on stochastic processes generated by dynamical systems. We develop a general framework for describing the dependence structure of extremes through two complementary objects: the tail process and the spectral measure. We first extend the enriched point-process convergence of the univariate theory to the multivariate setting, proving that the decorated point process of exceedances converges to a marked Poisson cluster process whose marks retain the componentwise and temporal structure of extreme clusters. We then construct a spectral measure for stationary sequences, which assigns weight to distinct extremal profiles and characterizes the asymptotic dependence between components, and establish a de Haan-Resnick type representation. We also relate the spectral measure to the tail process, showing that its normalized profile distribution is recovered from the angular component of the tail process. Finally, we recover the classical univariate and multivariate exceedance processes as special cases. We illustrate the theory through applications to dynamical systems, using examples in which different dynamical mechanisms give rise to distinct extremal clustering patterns and cross-dependence structures. These applications demonstrate the ability of the proposed framework to capture genuinely multivariate features of extreme events and to extract information about the underlying dynamics that is not accessible through classical exceedance processes.