The seminal papers of Pickands [1,2] paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian's Lemma, Borell-TIS and Piterbarg inequalities there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting obtaining the exact asymptotics of the exceedance probabilities for both stationary and non-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and the operator fractional Ornstein-Uhlenbeck process.

中文翻译：

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向量值高斯过程的极值

Pickands [1,2] 的开创性论文为系统研究平稳和非平稳高斯过程的高超越概率铺平了道路。然而，在向量值设置中，由于缺乏包括 Slepian 引理、Borell-TIS 和 Piterbarg 不等式在内的关键工具，文献中没有任何方法论发展来研究向量值高斯过程的极端情况。在这个贡献中，我们为向量值设置开发了统一双和方法，以获得平稳和非平稳高斯过程的超越概率的精确渐近线。我们将我们的发现应用于算子分数布朗运动和算子分数 Ornstein-Uhlenbeck 过程。