Extremes of $L^p$-norm of Vector-valued Gaussian processes with Trend

Abstract

Let $\mathbf{X}(t)=(X_1(t),\dots ,X_d(t))$ be a Gaussian vector process and $g(t)$ be a continuous function. The asymptotics of distribution of ${\Vert \mathbf{X}(t)\Vert }_p$, the $L^p$ norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of ${\Vert \mathbf{X}(t)\Vert }_p^c,c>0,$ with trend $g(t)$ over $[0,T]$. Both scenarios that $\mathbf{X}(t)$ is locally stationary and non-stationary are considered. Important examples include ${\sum }_{i=1}^d\left|X_i(t)\right|+g(t)$ and chi-square processes with trend, i.e., ${\sum }_{i=1}^d{X}_i^2(t)+g(t)$. These results are of interest in applications in engineering, insurance and statistics, etc.