Extremes of multidimensional Gaussian processes

This paper considers extreme values attained by a centered, multidimensional Gaussian process X(t)=(X1(t),…,Xn(t))X(t)=(X1(t),…,Xn(t)) minus drift d(t)=(d1(t),…,dn(t))d(t)=(d1(t),…,dn(t)), on an arbitrary set TT. Under mild regularity conditions, we establish the asymptotics of logP(∃t∈T:⋂i=1n{Xi(t)−di(t)>qiu}), for positive thresholds qi>0qi>0, i=1,…,ni=1,…,n and u→∞u→∞. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.