Extrema of multi-dimensional Gaussian processes over random intervals

Abstract

This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $$P(u):=\mathbb{P}\left\{{\cap }_{i=1}^n\left(\underset{t\in [0,{\mathcal{T}}_i]}{\sup }(X_i(t)+c_{i}t)>a_{i}u\right)\right\},u\to \infty ,$$ where $X_i(t),t\ge 0$, $i=1,2,\cdots ,n,$ are independent centered Gaussian processes with stationary increments, $\mathcal{T}=({\mathcal{T}}_1,\cdots ,{\mathcal{T}}_n)$ is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$, $a_i>0$, $i=1,2,\cdots ,n$. Our result shows that the structure of the asymptotics of $P(u)$ is determined by the signs of the drifts $c_i$'s. We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.