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Is there a Berry–Esseen bound for operator norm of an average of independent random matrices?

Suppose $A_1, \dotsc, A_n$ are independent matrices with $\mathbb{E}[A_i] = I$ (the identity matrix). Is there a Berry–Esseen bound for properly normalized $\lVert\overline{A} - I\rVert_\text{op}$?

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  • $\begingroup$ Given how underdeveloped the field of Matrix Chernoff like bounds is, a Berry Esseen type result seems too strong to currently exist, unfortunately. What do you need to bound? Maybe a simpler type of bound will do. $\endgroup$
    – Thomas Dybdahl Ahle
    Commented Apr 17, 2020 at 22:52

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Check out this paper on Berry–Esseen inequalities for random vectors, maybe it will be useful:

Bentkus - On the dependence of the Berry–Esseen bound on dimension.

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