For a random dynamical system the Lyapunov exponent is defined as:
$$\lambda(x) = \lim_{n\to\infty} \sup \frac{1}{n}\log||A_n \cdots A_1||,$$
where $A_i$ are i.i.d. random matrices. Furstenberg-Kesten theorem states that this limit does exist (provided $\mathbb{E}\log||A||<\infty$).
My question is, what rate does convergence occur at? Or more specifically, is there an upper bound on the convergence rate for any $A_i$?