Local stationarity in the extremes
Résumé
We introduce a novel approach that bridges extreme value theory and nonstationary processes, with the aim of characterizing the concept of E-local stationarity. In contrast to the conventional literature on locally stationary processes, the proximity between a nonstationary process and its stationary approximation cannot be defined solely in terms of the Lp norm, due to the possible non-existence of the first moment. We therefore propose a new definition of an E-local stationary process that captures the tail probabilities of nonstationary multivariate time series. This new framework does not rely on finite first moments or assumptions about the full distribution, but instead focuses exclusively on the limiting behavior of the tails. We show that several widely used processes satisfy this definition, including heavy-tailed linear processes and stochastic volatility models. Furthermore, we develop estimators for the limiting measure of a multivariate object and for the extreme quantiles of a univariate object, and establish their statistical properties, including consistency and asymptotic normality. Finally, we demonstrate the practical relevance of the proposed theory through an application to real-world data, specifically the daily log returns of two European stock indices.
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