A matrix concentration inequality for products

We present a non-asymptotic concentration inequality for the random matrix product \begin{equation}\label{eq:Zn} Zn = \left(Id-\alpha Xn\right)\left(Id-\alpha X{n-1}\right)\cdots \left(Id-\alpha X1\right), \end{equation} where $\left{Xk \right}{k=1}^{+\infty}$ is a sequence of bounded independent random positive semidefinite matrices with common expectation $\mathbb{E}\left[Xk\right]=\Sigma$. Under these assumptions, we show that, for small enough positive $\alpha$, $Zn$ satisfies the concentration inequality \begin{equation}\label{eq:CTbound} \mathbb{P}\left(\left\Vert Zn-\mathbb{E}\left[Z_n\right]\right\Vert \geq t\right) \leq 2d^{2\cdot\exp\left(\frac{-t2}{\alpha} \sigma^2} \right) \quad \text{for all } t\geq 0, \end{equation} where $\sigma^2$ denotes a variance parameter.