Approximation to Distribution of Product of Random Variables Using Orthogonal Polynomials for Lognormal Density

TL;DR

This paper develops a method using orthogonal polynomials to approximate the distribution of the product of lognormal-related random variables, applicable to both independent and correlated cases, with demonstrated accuracy in specific fading scenarios.

Contribution

It introduces a closed-form expression for orthogonal polynomials associated with the lognormal density, enabling practical approximation of product distributions.

Findings

Accurate approximation for product of Nakagami-m variables.

Effective under small cross-correlations and light fading conditions.

Provides a computationally efficient alternative to exact distributions.

Abstract

We derive a closed-form expression for the orthogonal polynomials associated with the general lognormal density. The result can be utilized to construct easily computable approximations for probability density function of a product of random variables, when the considered variates are either independent or correlated. As an example, we have calculated the approximative distribution for the product of Nakagami-m variables. Simulations indicate that accuracy of the proposed approximation is good with small cross-correlations under light fading condition.