On extremes for Gaussian subordination
Shuyang Bai, Marie-Christine Düker
TL;DR
This paper addresses the extreme-value behavior of stationary Gaussian-subordinated processes, including long-memory cases. It develops a nonstandard point-process framework that allows the latent Gaussian covariance to decay slower than any polynomial, nearly matching the Berman-type condition, and extends the theory to multivariate and infinite-dimensional transforms via truncation. The authors establish a multivariate extremal limit for the maxima, characterized by a Gumbel-type exponent and a multivariate extremal index \theta(\tau), with explicit forms in finite-dimensional moving-maxima models driven by subordinated Gaussian innovations. A key conceptual contribution is showing that Gaussian subordinations induce extremal independence under non-unity canonical correlations and introducing m-extremal dependence to handle truncation and long-range effects. The results are illustrated by a limit theorem for multivariate maxima of moving maxima processes, driven by regularly varying innovations from subordinated Gaussian sources, and by establishing D'(u_n) under weak covariance decay assumptions.
Abstract
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.