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Extremes of multidimensional Gaussian processes. (English) Zbl 1214.60010
Let $$X=\{X(t):\,t\in T\}$$ be a centered $$\mathbb R$$-valued Gaussian process indexed by $$T\subset \mathbb R$$. There is quite a large body of literature on characterizing the probability $$P(\sup_{t\in T}X(t)>u)=P(\exists t\in T:\,X(t)>u)$$ as $$u$$ increases. The prototype known result is of the form $$\lim_{u\to\infty} u^{-2}\log(P(\sup_{t\in T}X(t)>u))=-(2\sup_{t\in T}E(X(t)^2))^{-1}$$. Considerably less attention has been paid to multidimensional counterparts.
This paper establishes asymptotics of $$\log(P(\exists t\in T:\,\bigcap_{i=1}^n\{X_i(t)-d_i(t)> q_iu\}))$$ for positive thresholds $$q_i>0$$, $$1\leq i\leq n$$, as $$u\to\infty$$, where $$X(t)=(X_1(t),\dots,X_n(t))$$ is a centered multidimensional Gaussian process, $$d_i(t)$$ are drift functions and $$T$$ is an arbitrary subset of $$\mathbb R^m$$, $$n,m\in\mathbb N$$. The results generalize and extend known results for the one- and two-dimensional case.

##### MSC:
 60G15 Gaussian processes 60G70 Extreme value theory; extremal stochastic processes
##### Keywords:
Gaussian process; logarithmic asymptotics; extremes; exceedances
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