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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?

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    $\begingroup$ Have you investigated the case when each $A_i$ is diagonal? What if the matrices are symmetric and commute? $\endgroup$ Jun 10, 2020 at 21:53
  • $\begingroup$ @DieterKadelka I edited my post. We exclude the case where they commute, as that case would be perhaps an easier problem. $\endgroup$ Jun 10, 2020 at 22:03
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    $\begingroup$ For nonasymptotic bounds I have a recent preprint that might be useful: arxiv.org/pdf/2003.05437 $\endgroup$
    – J..
    Jun 11, 2020 at 1:45

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This is an extremely well studied problem. The canonical reference is Bougerol and LaCroix's book:

Bougerol, Philippe; Lacroix, Jean, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, Vol. 8. Boston - Basel - Stuttgart: Birkhäuser. X, 283 p. DM 88.00 (1985). ZBL0572.60001.

But the paper by Goldsheid-Margulis is also very good:

Gol’dshejd, I. Ya.; Margulis, G. A., Lyapunov exponents of a random matrix product, Usp. Mat. Nauk 44, No. 5(269), 13-60 (1989). ZBL0687.60008.

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