A matrix concentration inequality for products

Abstract

We present a non-asymptotic concentration inequality for the random matrix product \begin{equation}\label{eq:Zn} Z_n = \left(I_d-\alpha X_n\right)\left(I_d-\alpha X_{n-1}\right)\cdots \left(I_d-\alpha X_1\right), \end{equation} where $\left\{X_k \right\}_{k=1}^{+\infty}$ is a sequence of bounded independent random positive semidefinite matrices with common expectation $\mathbb{E}\left[X_k\right]=\Sigma$. Under these assumptions, we show that, for small enough positive $\alpha$, $Z_n$ satisfies the concentration inequality \begin{equation}\label{eq:CTbound} \mathbb{P}\left(\left\Vert Z_n-\mathbb{E}\left[Z_n\right]\right\Vert \geq t\right) \leq 2d^2\cdot\exp\left(\frac{-t^2}{\alpha \sigma^2} \right) \quad \text{for all } t\geq 0, \end{equation} where $\sigma^2$ denotes a variance parameter.