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?