Extremes of multidimensional Gaussian processes

This paper considers extreme values attained by a centered, 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 ( ∃ t ∈ T : ⋂ i = 1 n { X i ( t ) − d i ( t ) > q i u } ) , for positive thresholds q i > 0 , i = 1 , … , n and u → ∞ . Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.