Extremes of multidimensional Gaussian processes

Abstract

This paper considers extreme values attained by a centered, multidimensional Gaussian process $X(t)=(X_1(t),\dots ,X_n(t))$ minus drift $d(t)=(d_1(t),\dots ,d_n(t))$, 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 $q_i>0$, $i=1,\dots ,n$, 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.

Cited by in corpus (7)

• Bounds for expected supremum of fractional Brownian motion with drift
• Dominating Points of Gaussian Extremes
• Extremes of Vector-Valued Gaussian Processes
• Exact asymptotics of component-wise extrema of two-dimensional Brownian motion
• Logarithmic asymptotics for multidimensional extremes under non-linear scalings
• Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields
• Extremes of vector-valued Gaussian processes with Trend