Extremes of Gaussian chaos processes with Trend

TL;DR

This paper derives precise tail asymptotics for Gaussian chaos processes with trend functions, covering both stationary and non-stationary Gaussian vector processes, with applications to products of Gaussian processes and chi-processes.

Contribution

It provides the first comprehensive analysis of the tail behavior of Gaussian chaos processes with trend, including exact asymptotics for both stationary and non-stationary cases.

Findings

Derived exact tail asymptotics for Gaussian chaos processes with trend.

Analyzed both locally-stationary and non-stationary Gaussian processes.

Included applications to products of Gaussian processes and chi-processes.

Abstract

Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t)), t\in [0,S]$ be a Gaussian vector process and let $g(\boldsymbol{x}),\boldsymbol{x}\in\mathbb{R}^d$ be a continuous homogeneous function. In this paper we are concerned with the exact tail asymptotics of the chaos process $g(\boldsymbol{X}(t))+ h(t),t\in [0,S]$ with trend function $h$. Both scenarios $\boldsymbol{X}(t)$ is locally-stationary and $\boldsymbol{X}(t)$ is non-stationary are considered. Important examples include the product of Gaussian processes and chi-processes.