标题： 关于高斯次序的极值 显示英文标题

标题： On extremes for Gaussian subordination

摘要： 本文研究通过将变换应用于平稳高斯过程得到的过程的极值理论，也称为次高斯过程。 主要贡献如下。 首先，我们改进了\cite{sly2008nonstandard}的方法，以允许底层高斯过程的协方差衰减速度比任何多项式速率都慢，几乎符合伯曼的条件。 其次，我们将理论扩展到多变量设置中，其中次高斯过程和底层高斯过程都可以是向量值的，且变换是有限维的。 特别是，我们建立了从次高斯过程构造的点过程的弱收敛性，由此得出多变量极值极限定理。 一个有助于我们分析的关键观察结果，可能具有独立兴趣，如下所述：由两个具有非单位典型相关性的联合高斯向量的变换得到的任意二元随机向量始终保持极端独立。 这一观察也促使我们引入并讨论一个我们称之为 m-极端依赖的概念，它扩展了经典的 m-依赖概念。 此外，我们放松了对有限维变换的限制，通过近似论证将结果扩展到无限维设置。 作为示例，我们建立了一个由来自潜在长记忆的次高斯过程的规则变化创新驱动的多变量移动最大值过程的极限定理。 显示英文摘要

摘要： 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.