Consider $\boldsymbol X \sim \mathcal{N}(\boldsymbol 0, \boldsymbol \Sigma)$ and $\boldsymbol Y = (f_1(X_1), f_2(X_2),\dots, f_d(X_d))$. We call this a diagonal transformation of a multivariate normal. In this paper we compute exactly the mean vector and covariance matrix of the random vector $\boldsymbol Y.$ This is done two different ways: One approach uses a series expansion for the function $f_i$ and the other a transform method. We compute several examples, show how the covariance entries can be estimated, and compare the theoretical results with numerical ones.

翻译：考虑 $\boldsymbol X \sim \mathcal{N}(\boldsymbol 0, \boldsymbol \Sigma)$ 与 $\boldsymbol Y = (f_1(X_1), f_2(X_2),\dots, f_d(X_d))$。我们称之为多元正态分布的对角变换。本文精确计算了随机向量 $\boldsymbol Y$ 的均值向量与协方差矩阵。计算通过两种不同方法实现：一种方法使用函数 $f_i$ 的级数展开，另一种采用变换方法。我们计算了若干示例，展示了如何估计协方差矩阵的各个元素，并将理论结果与数值结果进行了比较。