This paper investigates extreme value theory for processes obtained by applying transformations to stationary Gaussian processes, also called subordinated Gaussian processes. The main contributions are as follows. First, we refine the method of \cite{sly2008nonstandard} to allow the covariance of the underlying Gaussian process to decay more slowly than any polynomial rate, nearly matching Berman's condition. Second, we extend the theory to a multivariate setting, where both the subordinated process and the underlying Gaussian process may be vector-valued, and the transformation is finite-dimensional. In particular, we establish the weak convergence of a point process constructed from the subordinated Gaussian process, from which a multivariate extreme value limit theorem follows. A key observation that facilitates our analysis, and may be of independent interest, is the following: any bivariate random vector derived from the transformations of two jointly Gaussian vectors with a non-unity canonical correlation always remains extremally independent. This observation also motivates us to introduce and discuss a notion we call m-extremal-dependence, which extends the classical concept of m-dependence. Moreover, we relax the restriction to finite-dimensional transforms, extending the results to infinite-dimensional settings via an approximation argument. As an illustration, we establish a limit theorem for a multivariate moving maxima process driven by regularly varying innovations that arise from subordinated Gaussian processes with potentially long memory.

翻译：本文研究通过对平稳高斯过程施加变换所获得过程的极值理论，这类过程亦称为从属高斯过程。主要贡献如下：首先，我们改进了\cite{sly2008nonstandard}的方法，使得基础高斯过程的协方差衰减速率可以慢于任何多项式速率，这几乎匹配了Berman条件。其次，我们将理论扩展至多元设定，其中从属过程与基础高斯过程均可为向量值，且变换为有限维。特别地，我们建立了由从属高斯过程构造的点过程的弱收敛性，由此导出一个多元极值极限定理。一个促进我们分析且可能具有独立意义的关键观察是：任何从两个具有非单位典型相关系数的联合高斯向量经变换导出的二元随机向量，总是保持极值独立性。这一观察也促使我们引入并讨论了一个称为m-极值依赖性的概念，该概念扩展了经典的m-依赖性概念。此外，我们放宽了对有限维变换的限制，通过近似论证将结果推广至无限维设定。作为例证，我们为多元滑动极大值过程建立了一个极限定理，该过程由正则变差创新驱动，这些创新源于可能具有长记忆的从属高斯过程。