Let random matrix $\mathbf{X} \in \mathbb{C^{\mathrm{m} \times \mathrm{n}}}$ and random vector $\mathbf{y} \in \mathbb{C^{\mathrm{m} \times 1}}$ are unknown distributed, but their covariance and correlation are known, $\mathbf{C}\_{\mathbf{X}}$, $\mathbf{C}\_{\mathbf{y}}$, and $\mathbf{C}\_{\mathbf{Xy}}$. The question is whether there is any bound for the following expression, (like Cauchy-Schwarz or Jensen's)


$?\le \mathbb{E}[(\mathbf{X}^{\mathrm{H}} \mathbf{X})^{-1} \mathbf{X}^{\mathrm{H}} \mathbf{y}] \le ?$


Also, I have to mention $\le$ here means an element-wise inequality since the corresponding expectation is an $\mathrm{n} \times 1$ vector.



Bound for expectation of random matrix
No, of course not.


Indeed, consider a simplest case when $m=n=1$, and $X:=\mathbf X$ and $y:=\mathbf y$ are iid standard normal random variables. Then
$$E(\mathbf{X}^{\mathrm{H}} \mathbf{X})^{-1} \mathbf{X}^{\mathrm{H}} \mathbf{y}=E\frac yX$$
does not even exist (because here $\frac yX$ has [the standard Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution#Properties)).


