Concentration inequality or a general bound on $\mathbb{E}[||\mathbb{E}[X]-X||]$

Question

Given a matrix random variable $X \in \mathbb{R}^{d \times d}$ is there a concentration or any interesting upper-bound on the following expression:

1. $ \mathbb{E}[||\mathbb{E}[X]-X||] \leq ??$

Information I have are spectral properties of the mean $\mathbb{E}[X]$.

2. Secondly, given that we have another estimator $S_n=\frac{1}{n} \sum_{i=1}^n X_i$ where $X_i$ (iid) like the random variable $X$ can I say something, about $ \mathbb{E}[||\mathbb{E}[S]-S||] $ where the expectation is on all $X_i$s.

The only idea I had first was to use Jensen's inequality with: $ \leq \mathbb{E}_{X,Y}[|| X - Y||]$, which leads again nowhere.

Answer 1

1. The most straightforward bound here is that the mean absolute deviation is bounded by the standard deviation.
2. In view of the bound given in 1, if $X_i$ are iid random variables with 2 moments and $\overline{X}_n=\frac{\sum_{i=1}^n X_i}{n}$, then $E[|E[X]-\overline{X}_n|]$ is bounded by the standard deviation of $\overline{X}_n$, which is $\frac{\sigma}{\sqrt{n}}$. In the particular case where $a \leq |X| \leq b$, $\sigma \leq \frac{b-a}{2}$.