I'd like to know if there is a concentration inequality for the sample covariance matrix that don't assume the knowledge of the true mean.

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**Background.**

Given a probability distribution $\mu$ on $\mathbb R^d$, the covariance matrix of $\mu$ is defined as follows:
$$\Sigma := \mathbb E [(x - \bar \mu)(x -\bar \mu)^\top] $$
where $x \sim \mu$ and $\bar \mu = \mathbb E [x]$. 

If $X = (x_1, \cdots x_m)$ is an i.i.d. sample drawn from $\mu$, then we can define two estimators:
\begin{align*}
& \hat \Sigma_1 := \frac1m \sum_{i=1}^m (x_i - \bar \mu)(x_i - \bar \mu)^\top, \text{ where } \bar \mu = \mathbb E_{x \sim \mu} [x] \\
& \hat \Sigma_2 := \frac1{m-1} \sum_{i=1}^m (x_i - \bar x)(x_i - \bar x)^\top, \text{ where } \bar x = \frac1m (x_1 + \cdots x_m)
\end{align*}
They both satisfy $\mathbb E_X \hat \Sigma_1 = \mathbb E_X \hat \Sigma_2 = \Sigma$.

The second estimator $\hat \Sigma_2$ is of interest because $\bar \mu$ is often not known in practice.

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**Question.**

I'm interested in the concentration of $\hat \Sigma_2$ to $\Sigma$ as $m \rightarrow \infty$. More precisely, given a number $t > 0$, I'd like to know whether there exists a constant $A>0$ and a term $\alpha \in (0,1)$ that depend on $\mu$ and $t$ such that 
$$\text{Prob}(\| \Sigma - \hat \Sigma_2 \| \ge t) \le A \cdot \alpha^m$$
where $\|\cdot \|$ is the spectral norm, also known as the 2-norm. (The Frobenius norm is also fine, since for any $d \times d$ matrix $A$, $\|A\| \le \|A\|_F \le \sqrt{d} \|A\|$)

In the case of the difference $\|\Sigma - \hat \Sigma_1\|$, such an answer can be obtained using the [matrix Bernstein inequality][1]. However, I'm less sure about $\|\Sigma - \hat \Sigma_2\|$. I have an idea, which is to use the fact that:
$$\hat \Sigma_1 - \hat \Sigma_2 = \frac1{m(m-1)} \sum_{i\neq j} (x_i-\bar\mu) (x_j-\bar\mu)^\top$$
which follows from:
\begin{align*}
 \hat \Sigma_2 =& \frac1m \sum_i x_i x_i^\top - \frac1{m(m-1)} \sum_{i\neq j} x_i x_j^\top \\
 =& \frac1m \sum_i (x_i-\bar\mu) (x_i-\bar\mu)^\top - \frac1{m(m-1)} \sum_{i\neq j} (x_i-\bar\mu) (x_j-\bar\mu)^\top \\
 =& \hat \Sigma_1 - \frac1{m(m-1)} \sum_{i\neq j} (x_i-\bar\mu) (x_j-\bar\mu)^\top
\end{align*}
But now I'm not sure how to control the sum of the quantities $(x_i-\bar\mu) (x_j-\bar\mu)^\top$, which are *not independent*. 

This should be a fairly standard question with a standard answer, but I couldn't find an answer to this. A [similar question][2]'s only answer wasn't addressing my question; it was addressing the case for $\hat \Sigma_1$.


  [1]: https://arxiv.org/abs/1501.01571
  [2]: https://math.stackexchange.com/questions/2374810/concentration-inequality-for-covariance