Convergence to Stable Laws and a Local Limit Theorem for Products of Positive Random Matrices

We consider the products $G_n=A_n\cdots A_1$ of independent and identical distributed nonnegative $d\times d$ matrices $(A_i{)}_{i\ge 1}$. For any starting point $x\in {\mathbb{R}}_{+}^d$ with unit norm, we establish the convergence to a stable law for the norm cocycle $\log \mathrm{\mid }{G}_{n}x\mathrm{\mid }$, jointly with its direction $G_n\cdot x=G_{n}x\mathrm{/}\mathrm{\mid }{G}_{n}x\mathrm{\mid }$. We also prove a local limit theorem for the couple $(\log \mathrm{\mid }{G}_{n}x\mathrm{\mid },G_n\cdot x)$, and find the exact rate of its convergence.