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Recently, I have found myself interested in concentration properties of random matrices.

Specifically I would like to answer questions of the following sort

Let $\{X_i\}_{i=1}^n$ be i.i.d. copies of a random symmetric positive definite matrix (with real entries), can we upper bound the quantity $\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^nX_i \ngeq \frac{1}{2}\mathbb{E}[X_1]\right)$?.

In the above, we take refer to the Loewner order on matrices, that is,for two matrices $A,B$ we say that $A \geq B$ if $A-B$ is a positive definite matrix.

However, most of the literature deals with concentration of the eigenvalues of random matrices.

Several relevant results can be found in Section 2.2.4.2 here. Specifically, Theorem 2.2.15 gives such a bound on the condition that $\mathbb{E}[X_1] \geq \mu\mathrm{I}$ for some $\mu > 0$

It seems to me that such a requirement is an artifact of the proof. It stands to reason that we could relax it.

My main inquiry is then, are there other such analogues for classic scalar inequalities (such as Bernstein's inequality) in the matrix setting? Ones which do not require that the relevant matrices be 'big' in a sense.

Since the exponent function is not matrix monotone, it is not true that whenever $A \geq B$ we have $e^A \geq e^B$. So, Chernoff's inequality need not not even apply in the matrix setting.

Instead, it would still be helpful to have an upper bound for $\mathbb{P}\left(e^{\frac{1}{n}\sum_{i=1}^nX_i} \nleq e^{\frac{1}{2}\mathbb{E}[X_1]}\right)$ directly.

Any references or known results will be welcome.

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