Proof of correct tending to infinity for a product of independent random matrices (Q1406362)

Let \(G\) be the group of unimodular matrices. The sequence of matrices \(g(n)\) correctly tends to infinity, if \[ \rho(g(n)) = \rho(k(n)) = \min\limits_{i=2,3} (t_i(n) - t_{i-1}(n))\to 0,\quad n\to \infty, \] with respect to probability. The main result of the paper is as follows. If the product \( g(n)=g_1g_2\dots g_n \) of the random independent matrices from \(G\) whose distribution has weakly symmetric measures \(\mu_i\) tends to infinity, then this tending to infinity is correct.