Asymptotic lyapunov exponents for large random matrices

Suppose that A1,..., AN are independent random matrices of size n whose entries are i.i.d. copies of a random variable . of mean zero and variance one. It is known from the late 1980s that when . is Gaussian then N.1 log . AN... A1 . converges to log $\sqrt{\mathrm{n}}$ as N . .. We will establish similar results for more general matrices with explicit rate of convergence. Our method relies on a simple interplay between additive structures and growth of matrices.