Asymptotics of finite system Lyapunov exponents for some random matrix ensembles

For products $P_N$ of $N$ random matrices of size $d \times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{\mu_i\}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements, and extended to the general variance case for $\mu_1$, methods known for the computation of $\lim_{N \to \infty} \langle \mu_i \rangle$ are used to compute the large $N$ form of the variances of the exponents. Analogous calculations are performed in the case that the matrices making up $P_N$ are products of sub-blocks of random unitary matrices with Haar measure. Furthermore, we make some remarks relating to the coincidence of the Lyapunov exponents and the stability exponents relating to the eigenvalues of $P_N$.