Asymptotic Lyapunov exponents for large random matrices

Abstract

Suppose that $A_{1},\dots ,A_{N}$ are independent random matrices of size $n$ whose entries are i.i.d. copies of a random variable $\xi $ of mean zero and variance one. It is known from the late 1980s that when $\xi $ is Gaussian then $N^{-1}\log \Vert A_{N}\dots A_{1}\Vert $ converges to $\log \sqrt{n}$ as $N\to \infty $. 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.