Excess and Deficiency of Extreme Multidimensional Random Fields

TL;DR

This paper derives explicit probability distributions for the excess and deficiency of energy in p-dimensional random vector fields, providing exact formulas and approximations validated by simulations.

Contribution

It introduces new explicit distributions for excess and deficiency in multidimensional random fields, with practical formulas for various thresholds.

Findings

Exact distributions derived for arbitrary thresholds

Simple approximate functions for high or low thresholds

Numerical simulations confirm analytical results

Abstract

Probability distributions and densities are derived for the excess and deficiency of the intensity or instantaneous energy (quasi-static power) associated with a $p$-dimensional random vector field. Explicit expressions for the exact distributions are obtained for arbitrary threshold levels, together with simple approximate functions for relatively high or low thresholds. It is shown that precise expressions only require an expansion of order $p-1$ in the ratio of the excess height to the threshold level. Numerical simulations validate the analytical results.