Extremes of multidimensional Gaussian processes

TL;DR

This paper derives asymptotic probabilities for the simultaneous extreme events of multidimensional Gaussian processes over arbitrary sets, extending known results from lower dimensions.

Contribution

It generalizes existing asymptotic results for Gaussian extremes from one- and two-dimensional cases to higher dimensions with arbitrary index sets.

Findings

Asymptotic formulas for joint tail probabilities of multidimensional Gaussian processes.

Extension of known results to arbitrary dimensions and sets.

Illustrative examples demonstrating the theory.

Abstract

This paper considers extreme values attained by a centered, multidimensional Gaussian process $X(t)= (X_1(t),\ldots,X_n(t))$ minus drift $d(t)=(d_1(t),\ldots,d_n(t))$, on an arbitrary set $T$. Under mild regularity conditions, we establish the asymptotics of \[\log\mathbb P\left(\exists{t\in T}:\bigcap_{i=1}^n\left\{X_i(t)-d_i(t)>q_iu\right\}\right),\] for positive thresholds $q_i>0$, $i=1,\ldots,n$, and $u\to\infty$. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.