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Product of random matrices which commute nearly certainly

In the paper “Matrix concentration for products” it’s said, that the next is simple to display.

Let $X_1,dots X_n$ breathe unbiased, bounded, sq. matrices, which commute nearly certainly. Define $Y_i=I+frac{X_i}n$. Then

$$ log mathbb{E}|Y_ndots Y_1|leq frac 1 n |sum_{i=1}^nmathbb{E}X_i| +Oleft(sqrt{frac{log d}n}privilege) $$

$|cdot|$ is the spectral norm and $d$ is the dimension of our matrices.

I do know the weaker inequality for matrices which don’t essentially commute nearly certainly with $sum_{i=1}^n|mathbb{E}X_i|$ as a substitute of $|sum_{i=1}^nmathbb{E}X_i|$.

Is this actually simple to display? How do I show it?

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