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

Jianzhang Mei, Quansheng Liu

TL;DR

This work studies products of i.i.d. nonnegative matrices by examining the log-norm cocycle S_n^x = log|G_n x| and the direction X_n^x = G_n·x. Using a transfer-operator framework with a spectral gap, the authors prove joint convergence of (S_n^x/a_n − b_n, X_n^x) to an α-stable law coupled with the stationary direction and establish a local limit theorem with an explicit rate. They derive exact convergence rates in law under refined tail and regular variation conditions, distinguishing two regimes based on ρ relative to −α, and introducing Δ and N-s terms to capture higher-order corrections. The results extend one-dimensional stable-limit theory to multivariate random matrix products, with potential applications to branching processes and random environments in high dimensions.

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.

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

TL;DR

This work studies products of i.i.d. nonnegative matrices by examining the log-norm cocycle S_n^x = log|G_n x| and the direction X_n^x = G_n·x. Using a transfer-operator framework with a spectral gap, the authors prove joint convergence of (S_n^x/a_n − b_n, X_n^x) to an α-stable law coupled with the stationary direction and establish a local limit theorem with an explicit rate. They derive exact convergence rates in law under refined tail and regular variation conditions, distinguishing two regimes based on ρ relative to −α, and introducing Δ and N-s terms to capture higher-order corrections. The results extend one-dimensional stable-limit theory to multivariate random matrix products, with potential applications to branching processes and random environments in high dimensions.

Abstract

We consider the products of independent and identical distributed nonnegative matrices . For any starting point with unit norm, we establish the convergence to a stable law for the norm cocycle , jointly with its direction . We also prove a local limit theorem for the couple , and find the exact rate of its convergence.
Paper Structure (8 sections, 12 theorems, 129 equations)

This paper contains 8 sections, 12 theorems, 129 equations.

Table of Contents

  1. Introduction and main results
  2. Convergence to stable laws and a local limit theorem
  3. Exact rate of convergence
  4. The transfer operator $P_t$ and the proof of Theorem \ref{['thm::local_limit_theorem']}
  5. The proof of Theorem \ref{['thm::main']}
  6. Auxiliary lemmas
  7. Proof of Theorem \ref{['thm::main']} for the case $\rho > -\alpha$
  8. Proof of Theorem \ref{['thm::main']} for the case $\rho < -\alpha$

Key Result

Theorem 1.1

Assume Conditions cond::allowability_and_positivity and cond::hennion. Then, there exist two sequences of real numbers $(a_n), (b_n)$, with $a_n \geq 0$ and $\lim_{n \to \infty}a_n = \infty$, and an $\alpha$-stable law $s_\alpha$, such that for any $x \in \mathbb{S}^{d-1}_+$, as $n \to \infty$, Moreover, if additionally $\alpha \neq 2$ and $\mu$ is non-arithmetic, then for any continuous function

Theorems & Definitions (26)

  • Theorem 1.1: Convergence to stable laws and local limit theorem
  • Remark 1
  • Theorem 1.2: Exact rate of convergence in law for $(S_n^x, X_n^x)$ with suitable norming
  • Proposition 2.1: hennion2008stable
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm::local_limit_theorem']}
  • ...and 16 more