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

TL;DR

This paper proves that the logarithm of the norm of products of positive random matrices converges to a stable law, and establishes a local limit theorem with explicit convergence rates for the joint distribution of the norm and direction.

Contribution

It introduces a new stable law convergence and a local limit theorem for products of positive random matrices, including explicit convergence rates.

Findings

Convergence to a stable law for the norm cocycle.

A local limit theorem with exact convergence rate.

Joint distribution convergence of norm and direction.

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.