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

Jianzhang Mei; Quansheng Liu

paper

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

arXiv:2505.07626

Abstract

We consider the products $G_{n} = A_{n} \dots A_{1}$ of independent and identical distributed nonnegative $d \times d$ matrices $(A_{i})_{i \geq 1}$ . For any starting point $x \in R_{+}^{d}$ with unit norm, we establish the convergence to a stable law for the norm cocycle $lo g ∣ G_{n} x ∣$ , jointly with its direction $G_{n} \cdot x = G_{n} x /∣ G_{n} x ∣$ . We also prove a local limit theorem for the couple $(lo g ∣ G_{n} x ∣, G_{n} \cdot x)$ , and find the exact rate of its convergence.