Extrema of multi-dimensional Gaussian processes over random intervals

TL;DR

This paper derives the asymptotic behavior of the joint tail probabilities of multi-dimensional Gaussian processes over random intervals, revealing the influence of drift signs and applying results to ruin probabilities in regenerative models.

Contribution

It provides the first detailed asymptotic analysis of multi-dimensional Gaussian process extrema over random intervals with varying drifts and random lengths.

Findings

Asymptotics depend on the signs of the drifts c_i.

Results characterize the tail behavior of joint maxima over random intervals.

Application to ruin probabilities in regenerative models.

Abstract

This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $$ P(u):=\mathbb{P}\left\{\cap_{i=1}^n \left(\sup_{t\in[0,\mathcal{T}_i]} ( X_{i}(t) +c_i t )>a_i u \right)\right\}, \ \ \ u\to\infty, $$ where $X_i(t), t\ge0$, $i=1,2,\cdots,n,$ are independent centered Gaussian processes with stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \cdots, \mathcal{T}_n)$ 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.