# Product of random matrices which commute almost surely

In the paper "Matrix concentration for products" it is stated, that the following is easy to show. Let $$X_1,\dots X_n$$ be independent, bounded, square matrices, which commute almost surely. Define $$Y_i=I+\frac{X_i}n$$. Then $$\log \mathbb{E}\|Y_n\dots Y_1\|\leq \frac 1 n \|\sum_{i=1}^n\mathbb{E}X_i\| +O\left(\sqrt{\frac{\log d}n}\right)$$ $$\|\cdot\|$$ is the spectral norm and $$d$$ is the dimension of our matrices.
I know the weaker inequality for matrices which do not necessarily commute almost surely with $$\sum_{i=1}^n\|\mathbb{E}X_i\|$$ instead of $$\|\sum_{i=1}^n\mathbb{E}X_i\|$$.
Is this really easy to show? How do I prove it?

• I am having a hard time understanding what the $\log d$ term in the error means. Is this saying that there is a universal constant, valid for any distribution of $X$ so the estimate applies? This seems spectacularly unlikely, especially if the $X$’s are of mean zero. If the constant depends on$X$, then so does $d$, so why write it? Jan 10 at 15:53
• I can give you the exact form of the mentioned weaker inequality. We call the bound for each $\|X_i\| \ L$. Then we get $$\|X_i/n+\mathbb{E}X_i/n\| \leq L/n+\|\mathbb{E}X_i\|/n=:\sigma_i$$ With $\upsilon =\sum_{i=1}^n\sigma_i^2$ we get $$\log \mathbb{E} \|Y_n\dots Y_1\|\leq 1/n \sum_{i=1}^n \|\mathbb{E}X_i\|+\sqrt{2\upsilon(2\upsilon \lor \log d)}$$ The observation $\upsilon\leq (2L)^2/n$ leads to the weaker result. Jan 10 at 16:37

It's true if the $$X_i$$'s are Hermitian. Then with probability 1 the matrices are simultaneously diagonalizable, so we may as well write everything in a basis where they are diagonal.
Then $$\mathbb E \|Y_n \dots, Y_n\| = \mathbb E \max_{j \in [d]} |(Y_n)_{jj} \dots (Y_1)_{jj}| = \mathbb E \max_{j \in [d]} \left|\prod_{i=1}^n \left(1 + \frac{(X_i)_{jj}}{n}\right) \right|.$$
If $$|(X_i)_{jj}| \leq C$$ almost surely, then each term is positive so it suffices to bound $$\log \mathbb E \max_{j \in [d]} e^{\sum_{i=1}^n (X_i)_{jj}/n}$$. Now, this quantity is less than
$$\log \mathbb E \max_{j \in [d]} e^{\sum_{i=1}^n ((X_i)_{jj} - \mathbb E (X_i)_{jj})/n} + \max_{j \in [d]} \frac 1n \mathbb E \sum_{i=1}^n (X_i)_{jj}.$$
The second term is bounded by $$\frac 1n \left\|\mathbb E \sum_{i=1}^n X_i \right\|$$. The first term is $$O(\sqrt{\log d/n})$$ by standard scalar concentration bounds for bounded random variables.