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.