The record method for two and three dimensional parameters random fields

TL;DR

This paper extends the record method for bounding the tail distribution of maxima in Gaussian random fields from two to three dimensions, providing new formulas and comparisons with existing techniques.

Contribution

It introduces new forms of the record method for 2D and extends it to 3D Gaussian fields, utilizing results on quadratic forms for improved bounds.

Findings

New formulas for the record method in 2D

Extension of the method to 3D Gaussian fields

Comparison showing advantages over other methods

Abstract

Let $S$ be a regular set of $\R^d$ and $X : S\rightarrow \R$ be Gaussian field with regular paths. In order to give bound to the tail of the distribution of the maximum, we use the record method of Mercadier. We present some new form in dimension 2 and extend it to dimension 3 using the result of the expectation of the absolute value of quadratic forms by Li and Wei. Comparison with other methods is conducted.