Not all positive-definite matrices are valid covariance matrices for lognormal variables

Question

A simple method to generate correlated lognormal variables $X_i$ that obey a covariance matrix $C_{\mathrm{ln}}$ with elements $c_{\mathrm{ln}}^{ij}$ is to first compute the covariance matrix $C_{\mathrm{g}}$ for the associated Gaussian variables $\ln(X_i)$, where the elements $c_{\mathrm{g}}^{ij}$ of $C_{\mathrm{g}}$ are given by:

$c_{\mathrm{g}}^{ij} = \ln \left(\frac{c_{\mathrm{ln}}^{ij}}{\langle X_i\rangle\langle X_j\rangle}+1\right)$,

where $\langle X_i \rangle$ is the mean of $X_i$. Then, we use $C_{\mathrm{g}}$ to generate correlated Gaussian variables that are later exponentiated to return $X_i$.

My question is: why some positive-definite $C_{\mathrm{ln}}$ result in non-positive-definite $C_{\mathrm{g}}$? For a $2\times2$ covariance matrix the answer is here, but even when such restrictions are satisfied the resulting $C_{\mathrm{g}}$ might still be non-positive-definite for $N\times N$ matrices with $N>2$.

Answer 1

Not all positive-definite $C_{ln}$ give valid $C_g$ because some covariance matrices are impossible to reach in this setup, because of the non-linearity between the original random variable and the variable of interest.

For example, if you considered the following problem: $X_1$ and $X_2$ and gaussian random variables of mean 0, possibly correlated, and $Y_1 = X_1$ and $Y_2 = (X_2)^2$ Then $Y_1$ and $Y_2$ would always be uncorrelated, no matter the original correlation between $X_1$ and $X_2$

A similar phenomenon is taking place for the log-normal tuning-curves. Because you go through the exponential non-linearity, some correlation values become impossible to reach