# concentration inequality for product of matrices

Suppose that we have $$N$$, $$n\times n$$ positive semidefinite matrices $$A_1, \cdots,A_N$$ such that $$0\preceq A_i \preceq I_n$$, $$\forall i$$. Also let us assume that $$A_i\neq I_n$$ for all $$i$$ and they are not commutable. Now we construct a sequence $$X(t)$$ as follows: At each iteration, we choose a matrix $$A(t)$$ from $$\{A_1,\cdots,A_N\}$$ uniformly at random and let $$X(t)=A(t)*X(t-1)$$. We also let $$X(0)=I_n$$. Let us also assume that $$X(t)\rightarrow 0$$ with probability 1 as $$t\rightarrow \infty$$. Then the question is whether we can compute bounds for the following $$\begin{equation} \mathbb{P}\left(\Vert X(t)-\mathbb{E}[X(t)]\Vert> \delta \right). \end{equation}$$

I don't expect any answer for the general case, but do we know results for specific cases?

• Have you investigated the case when each $A_i$ is diagonal? What if the matrices are symmetric and commute? Jun 10 '20 at 21:53
• @DieterKadelka I edited my post. We exclude the case where they commute, as that case would be perhaps an easier problem. Jun 10 '20 at 22:03
• For nonasymptotic bounds I have a recent preprint that might be useful: arxiv.org/pdf/2003.05437
– J..
Jun 11 '20 at 1:45