Extremes of multidimensional Gaussian processes

This paper considers extreme values attained by a centered, multidimensional Gaussian process 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 the logarithm of the probability that for some t in T, we have that (for all i = 1, ..., n) X_i(t) - d_i(t) > q_i u, for positive thresholds q_i > 0 and u large. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional case. A number of examples illustrate the theory.