The Variance-Gamma Product Distribution.

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.