Let $A,B$ be linear transformations in $\mathbb C^d$ such that $|\det(AB)|<1$ and have no non-trivial invariant subspaces in common. That is, if $V$ is an invariant subspace of $A$ and $W$ is of $B$ then $V\cap W$ is either $\{0\}$ or $\mathbb C^d$. Let $X_i$ be independent random matrices defined by
$$\mathbb P[X_i=A]=\mathbb P[X_i=B]=\frac 12.$$
Is it true that
$$\prod_{i=1}^n X_i\to 0$$
almost surely as $n\to\infty$?
This question is inspired by work on random dynamical systems. Linear functions are usually the simplest, and I couldn't find anything about random linear dynamical systems.
This problem has been studied. See for example:
Bellman, Richard, Limit theorems for non-commutative operators. I, Duke Math. J. 21, 491-500 (1954). ZBL0057.11202.
Furstenberg, H.; Kesten, H., Products of random matrices, Ann. Math. Stat. 31, 457-469 (1960). ZBL0137.35501.
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.
Pollicott, M., Effective estimates of Lyapunov exponents for random products of positive matrices, ZBL07384583.