Extremes of vector-valued Gaussian processes: exact asymptotics

TL;DR

This paper derives exact asymptotics for the probability that all components of a vector-valued Gaussian process exceed a high threshold over a time interval, extending extremal theory with new inequalities and constants.

Contribution

It provides the first precise asymptotic formulas for multivariate Gaussian processes exceeding thresholds, along with extensions of key inequalities and constants.

Findings

Exact asymptotics for vector Gaussian process maxima

Extensions of Piterbarg, Borell-TIS, Slepian, and Pickands-Piterbarg inequalities

Analysis of multidimensional Pickands and Piterbarg constants

Abstract

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$ as $u\to\infty$, for both locally stationary $X_i$'s and $X_i$'s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.