Extremal ranks and transformation of variables for extremes of functions of multivariate Gaussian processes

The exit rate from a ‘safe region’ plays an important role in dynamic reliability theory with multivariate random loads. For Gaussian processes the exit rate is simply calculated only for spherical or linear boundaries. However, many smooth boundaries, not of any of these types, are asymptotically spherical in variables of lower dimension, having a greater curvature in the remaining variables. As is shown in this paper, the asymptotic exit rate is then simply expressed as the exit rate from a sphere for a process of the lower dimensions, corrected by an explicit factor.The procedure circumvents the need to calculate complicated exit rate integrals for general boundaries, reducing the problem to a Gaussian probability integral for independent variables.A result of independent interest relates the tail distribution for a sum of a noncentral χ²-variable and a weighted sum of squares of noncentral normal variables, to the tail distribution of the χ²-variable only.