Let $\boldsymbol{X}\in\mathbb{R}^n$ be a random variable with positive entries ($X_i\geq a>0$). I want to characterize the relation between the second moment matrix $\boldsymbol{M}$, defined as

$$ \boldsymbol{M} = \mathbb{E}\{\boldsymbol{X}\boldsymbol{X}^*\}, $$

and that we can assume to be positive definite, and the matrix

$$ \boldsymbol{N} = \mathbb{E}\{\log(\boldsymbol{X})\boldsymbol{X}^*\}, $$

where $\log(\boldsymbol{X})$ is point-wise logarithm of the entries of $\boldsymbol{X}$. In particular, I want to characterize the product of $\boldsymbol{N}$ with $\boldsymbol{M}^{-1}$:

$$ \boldsymbol{N} \boldsymbol{M}^{-1} = \color{red}{\quad ?} $$

Experimentally, by computing these statistics on my data, I find that this product appears to be equal to the sum of a multiple of the identity, plus a matrix with constant columns:

$$ \boldsymbol{N} \boldsymbol{M}^{-1} \ \stackrel{?}{\simeq}\ \alpha \boldsymbol{I}+ \boldsymbol{1}\boldsymbol{v}^* $$ Moreover, this matrix seems to be diagonally dominant.

I tried to play around with other functions other than $\log$ and it looks that this relation holds for other monotone functions as well.

I tried several approaches but I did not did not get anything out of them (Taylor expansions, copulas, etc.) though this might be due to my not particularly advanced graduate level analysis skills.

There are some simple transformation to the problems that do not seem to lead anywhere --- for example, $\boldsymbol{M} = \text{cov}(\boldsymbol{X},\boldsymbol{X}) + \boldsymbol{\mu}\boldsymbol{\mu}^*$, and so $\boldsymbol{M}^{-1}$ can be written using the Sherman–Morrison formula for a rank-one update: $\boldsymbol{M}^{-1} = \text{cov}(\boldsymbol{X},\boldsymbol{X})^{-1} - \boldsymbol{a}\boldsymbol{a}^*$ for some vector $\boldsymbol{a}^*$.

Eventually one arrives at estimating the product

$$ \text{cov}(\log(\boldsymbol{X}),\boldsymbol{X}) \text{cov}(\boldsymbol{X},\boldsymbol{X})^{-1} = \color{red}{\quad ?} $$

which I guess is the crux of the matter...

**update**: The Bussgang Theorem says that for **stationary** and **Gaussian** processes, we have:

$$ \text{corr}(\mathbf{X},f(\mathbf{X})) = C_f \text{corr}(\mathbf{X},\mathbf{X}) $$

where $C_f$ is a constant that depends only on $f$, which, if you ask me, it is pretty surprising.