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

Abstract

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