Some Limit Theorems Regarding Products of Random Matrices I: Directional Derivative of the Lyapunov Exponent

Given an i.i.d. sequence $\{A_n(\omega)\}_{n\ge 1}$ of invertible matrices and a random matrix $B(\omega)$, we consider the random matrix sequences inductively defined by $S_n(\omega) = A_n(\omega)S_{n-1}(\omega)$ and $T_n(\omega) = B(\sigma^{n-1}\omega)S_{n-1}(\omega)+A_n(\omega)T_{n-1}(\omega)$, and study several limit theorems involving $T_n(\omega)$ as well as the asymptotic behaviour of the action of $T_n(\omega)$ on the projective space and on the unit circle.