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

TL;DR

This paper extends the volume-of-tube method to Gaussian random fields with inhomogeneous variance, providing new formulas for tail probability approximations and analyzing the asymptotic errors involved.

Contribution

It generalizes the tube method to non-constant variance cases, deriving volume and tail probability formulas and analyzing approximation errors.

Findings

Derived volume formula for spherical tubes with non-constant radius

Provided tail probability approximation for Gaussian fields with inhomogeneous variance

Analyzed asymptotic approximation error and critical radius generalization

Abstract

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