零均值相关正态随机变量乘积分布的基本理论
The basic distributional theory for the product of zero mean correlated normal random variables
摘要：
两个零均值相关正态随机变量的乘积，更一般地说是这类随机变量的独立拷贝之和，在统计学文献中受到了广泛的关注，并出现在许多应用领域。然而，许多重要的分布特性尚未被记录。本文填补了这一空白，提供了基本的分布理论的两个零均值相关的正态随机变量的产品的独立副本的总和。所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，模式和中位数，Stein特征，其他随机变量的表示，以及相关分布的列表。我们还回顾了两个零均值相关的正态随机变量的乘积如何自然地作为一个极限分布出现，并给出了一个双Wiener-Itô积分的分布近似的例子。
英文摘要：
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.
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