On the product of correlated normal random variables and the noncentral chi-square difference distribution

2024/08/07 by Robert E. Gaunt, Gaunt, Robert E.
Computer Science · #62E15 #Bayesian Methods and Mixture Models #FOS: Mathematics #Primary 60E05 #Probability (math.PR)

paper · doi:10.48550/arxiv.2408.04101

openalex publication_date 2024/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We represent the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, as a difference of two independent noncentral chi-square random variables (which we refer to as the noncentral chi-square difference distribution). As a consequence, we obtain, amongst other results, an exact formula for the probability density function of the noncentral chi-square difference distribution, a Stein characterisation of the noncentral chi-square difference distribution, a simple formula for the moments of the sum of independent copies of the product of correlated normal random variables, an exact formula for the probability that such a random variable is negative, and also show that such random variables are self-decomposable and provide a Lévy-Khintchine representation of the characteristic function.

Related

• A Stein characterisation of the distribution of the product of correlated normal random variables
• The basic distributional theory for the product of zero mean correlated normal random variables
• On mixture relationships between central and non-central chi-squared difference distributions
• Moments of the log non-central chi-square distribution
• Infinite Divisibility of the Product of Two Correlated Normal Random Variables and Exact Distribution of the Sample Mean