Exact mean and covariance formulas after diagonal transformations of a multivariate normal

Consider $\mathit{X}\sim \mathcal{N}(\mathbf{0},\mathbf{\Sigma })$ and $\mathit{Y}=(f_1(X_1),f_2(X_2),\dots ,f_d(X_d))$. 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 $\mathit{Y}\mathrm{.}$ 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.