Stein’s method and the distribution of the product of zero mean correlated normal random variables

Abstract

Over the last 80 years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables. Motivated by this historical interest, we use a recent technique from the Stein's method literature to obtain a simple new proof, which also serves as an exposition of a general method that may be useful in related problems.

Phenotypic distribution and selection response for linear-product gene-environment interaction in quantitative traits

“…In recent years, an exact form for the distribution of the product of two normal variables has been derived by (Nadarajah and Pogány, 2016;Cui et. al., 2016) and also been related to the variance-gamma distribution (Gaunt, 2014(Gaunt, , 2019(Gaunt, , 2021. In particular, for two correlated normal distributions with mean 0, variances σ 2 G and σ 2 E , and geneenvironment correlation r GXE the probability density of their product I = GE is…”

“…The term K 0 in the two equations represents the modified Bessel function of the second kind of order zero. While not a widely familiar function, the moments for this distribution can be stated below by reference to the related variance-gamma distribution (Gaunt, 2014(Gaunt, , 2019(Gaunt, , 2021. Variance-gamma (V G) distributions are described in the appendix, but the distribution f (I)…”

“…In recent years, an exact form for the distribution of the product of two normal variables has been derived by (Nadarajah and Pogány, 2016; Cui et. al., 2016) and also been related to the variance-gamma distribution (Gaunt, 2014, 2019, 2021). In particular, for two correlated normal distributions with mean 0, variances and , and gene-environment correlation r GXE the probability density of their product I = GE is In the case where there is no GXE covariance and r GXE = 0, this reduces to The term K 0 in the two equations represents the modified Bessel function of the second kind of order zero.…”

“…In particular, for two correlated normal distributions with mean 0, variances and , and gene-environment correlation r GXE the probability density of their product I = GE is In the case where there is no GXE covariance and r GXE = 0, this reduces to The term K 0 in the two equations represents the modified Bessel function of the second kind of order zero. While not a widely familiar function, the moments for this distribution can be stated below by reference to the related variance-gamma distribution (Gaunt, 2014, 2019, 2021). Variance-gamma ( V G ) distributions are in the appendix, but the distribution The first and second moments of this distribution are given by Again, when GXE covariance is absent These moments lead to several conclusions about the effect of linear-product gene-environment interaction on the phenotypes of quantitative traits.…”

Approximating the Distribution of the Product of Two Normally Distributed Random Variables

“…For example, the Bessel function method proposed in [9] presents a good approximation, but only for the product of two standard normal distributions and, moreover, it is not defined at zero. In [12][13][14][15], other approaches are presented, including Stein's operators and distributional theory. Another approach is to use any other distribution to approximate the probability density function (PDF) of the mentioned product.…”