Extremes of multidimensional Gaussian processes

This paper considers extreme values attained by a cemered, multidimensional Gaussian process X(t) = (X-1(t), ... , X-n (t)) minus drift d(t) = (d(1) (t), ... , d(n) (t)), on an arbitrary set T. Under mild regularity conditions, we establish the asymptotics of log P (there exists t is an element of T : boolean AND(n)(i=1) {x(i) (t) - d(i) (t), q(i)u}), for positive thresholds q(i) > 0, i = l, ... , n and u -> infinity. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases A number of examples illustrate the theory. (C) 2010 Elsevier B.V. All rights reserved.