On computing distributions of products of random variables via Gaussian multiresolution analysis

TL;DR

This paper introduces Gaussian multiresolution analysis (GMRA) for efficiently approximating the distribution of products of independent random variables, outperforming Monte Carlo methods in accuracy and enabling Gaussian mixture representations.

Contribution

The paper presents a novel Gaussian MRA framework that allows accurate PDF computation of products of random variables, including an exact MRA construction for specified accuracy.

Findings

GMRA achieves higher accuracy than Monte Carlo methods.

The resulting PDF is expressed as a Gaussian mixture.

Exact MRA can be constructed for user-defined accuracy.

Abstract

We introduce a new approximate multiresolution analysis (MRA) using a single Gaussian as the scaling function, which we call Gaussian MRA (GMRA). As an initial application, we employ this new tool to accurately and efficiently compute the probability density function (PDF) of the product of independent random variables. In contrast with Monte-Carlo (MC) type methods (the only other universal approach known to address this problem), our method not only achieves accuracies beyond the reach of MC but also produces a PDF expressed as a Gaussian mixture, thus allowing for further efficient computations. We also show that an exact MRA corresponding to our GMRA can be constructed for a matching user-selected accuracy.