A diffusion equation for the density of the ratio of two jointly distributed Gaussian variables and the numerical inversion of Laplace transform

TL;DR

This paper derives a diffusion equation for the density of the ratio of two joint Gaussian variables with the same variance, and explores its implications for kernel density estimation and numerical Laplace transform inversion.

Contribution

It introduces a novel diffusion equation governing the ratio of Gaussian variables and applies it to improve kernel density estimation in Laplace transform inversion.

Findings

Density of the ratio satisfies a non-stationary diffusion equation

Implications for kernel density estimation of generalized eigenvalues

Enhanced methods for numerical Laplace transform inversion

Abstract

It is shown that the density of the ratio of two random variables with the same variance and joint Gaussian density satisfies a non stationary diffusion equation. Implications of this result for kernel density estimation of the condensed density of the generalized eigenvalues of a random matrix pencil useful for the numerical inversion of the Laplace transform is discussed.