reference request – When is $ E left[f(X)X^{top}right]$ invertible?

Let $ mathcal {M} _ { mu} $ either the set of probability measures over the measurable space $ left ( mathcal {B}, mathbb {R} ^ N right) $ which are absolutely continuous with respect to Lebesgue's measure $ mu $ sure $ left ( mathcal {B}, mathbb {R} ^ N right) $, or $ mathcal {B} $ is the sigma-algebra of Borel on $ mathbb {R} ^ N $. Let $ m mathcal {B} $ the set of measurable maps of Borel $ f ,: , mathbb {R} ^ N rightarrow mathbb {R} ^ N $.

Set the set

$ mathcal {C} overset { Delta} = left {(f, nu) in m mathcal {B} times mathcal {M} _ { mu} ,: , det left (E left[f(X)X^{top}right] right) neq 0, , X sim nu, , E left[left|left|f(X)X^{top}right|right|right]< infty right } $.

or $ X sim nu $ means that $ X $ is a random vector with a probability distribution (or a push-forward measure) $ nu $; $ X ^ { top} $ represents the transposition of $ X $; and $ left | left | cdot right | right | $ is a matrix standard.

Question 1. Is there a reference (literature) on the characterization of $ mathcal {C} $? In other words, is there a reference (or set of references) that explores the conditions of the pair? $ left (f, nu right) $ this invertability of performance of $ E left[f(X)X^{top}right]$? Any idea about the structure of this set would be appreciated.


What I know. [you may skip this part]

$ bullet $ It's easy to show that $ left ({ id sf}, nu right) in mathcal {C} $ for all $ nu in mathcal {M} _ { mu} $, or $ { sf id} $ is the identity card. In other words, $ E left[X X^{top}right]$ is invertible each time $ X $ is absolutely continuous.

$ bullet $ It's easy to show that if $ f $ is linear of full rank, that is, $ f (y) = Ay $with $ A in mathbb {R} ^ {N times N} $ an invertible matrix and then $ (f, nu) in mathcal {C} $ for everyone $ nu in mathcal {M} _ { mu} $. In other words, $ det left (E left[AXX^{top}right] right) neq 0 $ does not matter when $ X $ is absolutely continuous and $ A $ is invertible.

$ bullet $ With a little more effort, we can show that for almost all piecewise linear maps $ f $ (in a formal sense that I'm going to jump here), we have $ left (f, nu right) in mathcal {C} $ for everyone $ nu in mathcal {M} _ { mu} $.

$ bullet $ We can observe that if $ nu $ is a product of Gaussian (mean zero), that is to say $ X sim nu $ is Gaussian (multivariate) of zero mean with independent coordinates, and $ f (y) = left[y_1^k,,y_2^k,ldots,y_N^kright]^ { top} $ with k $ even then $ E left[f(X)X^{top}right]= $ 0 and therefore, not invertible.

The last example illustrates that canonical examples of $ f $ and $ nu $ – in this case, a polynomial function and a zero mean Gaussian distribution – may not allow the invariance of the covariance matrix $ E left[f(X)X^{top}right]$. At the same time, the last example seems a rather special case – for almost all Medium non-zero multivariate Gaussian matrix $ E left[f(X)X^{top}right]$ will be invertible, with the $ f $ defined in the last bullet.


Question 2. From the above discussion, my naive intuition is that $ E left[f(X)X^{top}right]$ is invertible for almost all elements in $ mathcal {C} $but it would be necessary to install an appropriate measure $ mathcal {C} $ first to prove / refute this vaguely stated statement. How to formalize and prove / refute this statement?


A little context. In the estimation of some parameters of a stochastic dynamic system through the observation of the evolution of the system, I have to reverse $ E left[f(X)X^{top}right]$, or $ f $ characterizes a part of the dynamic law and $ nu $ (the distribution of $ X $) is the limit distribution of the dynamic system. Therefore, I would like to understand the conditions on $ f $ and $ nu $ thanks to which I can invert this matrix to estimate the parameters.