The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener-Itô integrals.

中文翻译：

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零均值相关正态随机变量乘积的基本分布理论

两个零均值相关正态随机变量的乘积，更一般地说是这些随机变量的独立副本的总和，在统计学文献中受到了广泛关注，并出现在许多应用领域。然而，许多重要的分布特性还有待记录。这篇评论论文通过为两个零均值相关正态随机变量的乘积的独立副本之和提供基本分布理论来填补这一空白。涵盖的属性包括概率和累积分布函数、生成函数、矩和累积量、众数和中位数、Stein 表征、其他随机变量的表示以及相关分布的列表。