Extremes of vector-valued Gaussian processes with Trend

Abstract

Let $X(t)=(X_1(t),\dots ,X_n(t)),t\in \mathcal{T}\subset \mathbb{R}$ be a centered vector-valued Gaussian process with independent components and continuous trajectories, and $h(t)=(h_1(t),\dots ,h_n(t)),t\in \mathcal{T}$ be a vector-valued continuous function. We investigate the asymptotics of $$\mathbb{P}\left(\underset{t\in \mathcal{T}}{\sup }\underset{1\le i\le n}{\min }(X_i(t)+h_i(t))>u\right)$$ as $u\to \infty$. As an illustration to the derived results we analyze two important classes of $X(t)$: with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.

24 pages