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.