When does the product of random matrices diverge?

Question

Suppose $A_i$ are IID samples of a random matrix-valued variable. I'm interested in determining whether the following infinite product is likely to diverge

$$A_1 A_2 A_3\cdots$$

Finding necessary+sufficient condition is hard (Theorem 2).

• There are cheap to compute sufficient conditions for the product to converge.

• Are there cheap to compute sufficient conditions for the product to diverge?

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Of particular interest is case of $A_i=(I-x_i x_i^T)$ where $x_i$ are IID Gaussian. The case for isotropic Gaussian is solved here, I'm looking for insight into non-isotropic case

Answer 1

You generate a large enough sample of $x_i$ from a non-isotropic Gaussian distribution. Then for a sufficient number of samples n, generate matrices $A_1,A_2,\dots, A_n$. Then calculate the norm of product of these matrices. By Oseledets theorem, there is a $\lambda$ such that \begin{equation} \lim_{n \to \infty} \frac{1}{n} \log{(\lVert A_1 A_2 \cdots A_n \rVert)} = \lambda \end{equation} Where if $\lambda>0$, the product will almost surely diverge, while if $\lambda < 0$, the product will almost surely converge. For sufficient $n$, this can be estimated, which will provide reasonable parameters for the conditions you want. I don't think this answers the question in the way you were hoping unfortunately.