The volume-of-tube method for Gaussian random fields with inhomogeneous variance

The tube method or the volume-of-tube approximates tail probability of maximum a smooth Gaussian random field with zero mean and unit variance. This evaluates volume spherical about index set, then transforms it to probability. In this study, we generalize case in which variance is not constant. We provide formula for non-constant radius terms curvature tensors, inhomogeneous variance, as well its Laplace approximation. particular, critical generalized evaluation asymptotic approximation error. As an example, discuss largest eigenvalue distribution Wishart matrix non-identity parameter. Bonferroni when set finite set. error