# Matrix Azuma Inequality with non-zero mean

Based on theorem 7.1 of the paper User-friendly tail bounds for sums of random matrices, we have:

Consider a finite adapted $$\{ \mathbf{X}_k\}_{k=0}^{\infty}$$ of self-adjoint matrices in dimension $$d$$, and a fixed sequence $$\{ \mathbf{A}_k\}_{k=0}^{\infty}$$ of self-adjoint matrices that satisfy

$$\mathbb{E}[\mathbf{X}_k]=0 \quad \text{and } \quad \mathbf{X}_k\preceq \mathbf{A}_k \quad \text{almost surely}$$ Compute the variance parameter $$\sigma^2 = \| \sum_{k} \mathbf{A}_k^2 \|$$ Then, for all $$t\ge 0$$, $$\mathbb{P} \left\{ \lambda_{\max}\left( \sum_k\mathbf{X}_k \right)\ge t \right\}\le d e^{-\frac{t^2}{8\sigma^2}}$$

In this formula, the condition, $$\mathbb{E}[\mathbf{X}_k]=0$$ must hold. I am wondering if it is possible to extend it to the case that $$\mathbb{E}[\mathbf{X}_k]=\mu_k \neq 0$$? (Should $$\mu_k$$ be a matrix? So, it might be better to use sth like $$\Sigma_k$$, not $$\mu_k$$). Does expected value mean element-wise expectation?

If there is any references, I would be thankful if you could provide them.