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

Consider $X\sim N(0,\Sigma )$ and $Y=(f_1\left(X_1\right),f_2\left(X_2\right),\dots ,f_d\left(X_d\right))$. 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 $Y.$ This is done in 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.