Extremes of multidimensional Gaussian processes

Details Extremes of multidimensional Gaussian processes Extremes of multidimensional Gaussian processes

2010-05-31

arxiv.org/abs/1006.0029v3

Stochastic Processes and their Applications 120 (2010) 2289-2301

Mathematics

Probability

Scientific paper

10.1016/j.spa.2010.08.010

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 \[ \log\pp\left(\exists{t\in T}:\bigcap_{i=1}^n\left\{X_i(t)-d_i(t)>q_iu\right\}\right), \] for positive thresholds $q_i>0$, $i=1,\ldots,n$, and $u\toi$. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.

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