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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
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