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The Variance-Gamma Product Distribution.

Robert E Gaunt1, Siqi Li1, Heather L Sutcliffe1

  • 1Department of Mathematics, The University of Manchester, Oxford Road, M13 9PL Manchester, UK.

Results in Mathematics
|September 26, 2025
PubMed
Summary

This study provides the exact probability density function for the product of N independent variance-gamma random variables. These findings also yield formulas for related functions and approximations for specific probability distributions.

Keywords:
Laplace distributionMeijer G-functionNormal distributionProduct distributionVariance-gamma distribution

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Area of Science:

  • Probability theory
  • Mathematical statistics
  • Stochastic processes

Background:

  • The product of independent random variables is a fundamental concept in probability theory.
  • Variance-gamma, asymmetric Laplace, Laplace, and centered normal distributions are important in various statistical applications.
  • Deriving exact distributions for products of random variables can be analytically challenging.

Purpose of the Study:

  • To derive the exact probability density function (PDF) for the product of N independent variance-gamma random variables.
  • To extend these results to obtain formulas for the cumulative distribution function (CDF), characteristic function, and asymptotic approximations.
  • To deduce closed-form formulas for related distributions, including products of asymmetric Laplace, Laplace, and centered normal random variables.

Main Methods:

  • Exact derivation of the probability density function for the product of independent variance-gamma random variables.
  • Application of the derived PDF to obtain formulas for CDF and characteristic functions.
  • Development of asymptotic approximations for density, tail probabilities, and quantile functions.

Main Results:

  • An exact formula for the PDF of the product of N independent variance-gamma random variables with a zero location parameter.
  • Formulas for the CDF and characteristic function of this product.
  • Asymptotic approximations for density, tail probabilities, and quantile functions.
  • Closed-form formulas for the PDF, CDF, and characteristic function of products involving asymmetric Laplace, Laplace, and centered normal random variables.

Conclusions:

  • The study successfully derives the exact PDF for the product of independent variance-gamma random variables.
  • The derived methodology provides a framework for analyzing products of related probability distributions.
  • The results offer valuable tools for statistical modeling and analysis involving complex random variable products.