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.
Categories: math.PR stat.ML