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Convergence to Stable Laws for Products of Random Matrices

Axel Péneau

TL;DR

This work analyzes the log-norm κ(g)=log∥g∥ of products of i.i.d. random matrices γ_n ∼ ν in GL(E) under strong irreducibility and proximality, proving a generalized central limit theorem (GCLT) for κ(γ̄_n$ in the infinite-variance regime. The authors introduce a norm-cancellation quantity Δκ and establish sharp probabilistic bounds with squared tails, enabling a gain of moments via a pivoting construction that decomposes words into quasi-independent blocks. They extend these ideas to higher rank through Cartan projections, defining κ_i and the Cartan vector ḱappa and proving LLN and CLT for the corresponding Δκ and Δdotκ under appropriate irreducibility/proximality assumptions. The results provide a unified probabilistic framework, showing that the sum of log-norm contributions can converge to α-stable laws (α∈(0,2]) or Gaussian limits (α=2) depending on moment conditions, with a general pathway from LLN/CLT for almost additive processes to the main matrix-product conclusions.

Abstract

Under reasonable algebraic assumptions and under an infinite second order moment assumption, we show that the logarithm of the norm (log-norm) of a product of random i.i.d. matrices with entries in $\mathbb{R}$ or in any other local field satisfies a generalized Central Limit Theorem (GCLT) in the sense of Paul Lévi. The proof is based on a weak law of large number for the difference $Δ_n$ between the log-norm of the product of the first $n$ matrices and the sum of their log-norms. This weak law of large numbers morally says that $Δ_n$ behaves like a sum of i.i.d. random variables that have a finite moment of order $2q$ as long as the log-norm of each matrices has a finite moment of order $q$ for a given $q > 0$. This gain of moment is the central result of the present paper and is based on the construction of pivotal times. Moreover, these results admit a nice higher rank extension when one looks at the full Cartan projection instead of the log-norm.

Convergence to Stable Laws for Products of Random Matrices

TL;DR

This work analyzes the log-norm κ(g)=log∥g∥ of products of i.i.d. random matrices γ_n ∼ ν in GL(E) under strong irreducibility and proximality, proving a generalized central limit theorem (GCLT) for κ(γ̄_n$ in the infinite-variance regime. The authors introduce a norm-cancellation quantity Δκ and establish sharp probabilistic bounds with squared tails, enabling a gain of moments via a pivoting construction that decomposes words into quasi-independent blocks. They extend these ideas to higher rank through Cartan projections, defining κ_i and the Cartan vector ḱappa and proving LLN and CLT for the corresponding Δκ and Δdotκ under appropriate irreducibility/proximality assumptions. The results provide a unified probabilistic framework, showing that the sum of log-norm contributions can converge to α-stable laws (α∈(0,2]) or Gaussian limits (α=2) depending on moment conditions, with a general pathway from LLN/CLT for almost additive processes to the main matrix-product conclusions.

Abstract

Under reasonable algebraic assumptions and under an infinite second order moment assumption, we show that the logarithm of the norm (log-norm) of a product of random i.i.d. matrices with entries in or in any other local field satisfies a generalized Central Limit Theorem (GCLT) in the sense of Paul Lévi. The proof is based on a weak law of large number for the difference between the log-norm of the product of the first matrices and the sum of their log-norms. This weak law of large numbers morally says that behaves like a sum of i.i.d. random variables that have a finite moment of order as long as the log-norm of each matrices has a finite moment of order for a given . This gain of moment is the central result of the present paper and is based on the construction of pivotal times. Moreover, these results admit a nice higher rank extension when one looks at the full Cartan projection instead of the log-norm.
Paper Structure (15 sections, 31 theorems, 181 equations)

This paper contains 15 sections, 31 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Notations and motivation
  3. Probabilistic estimate for the norm-cancellation function
  4. The Generalized Central Limit Theorem (GCLT)
  5. GCLT in higher rank
  6. Pivoting technique and gain of moment
  7. Pivoting technique
  8. Study of the tail of the blocks
  9. Study of almost additive processes
  10. Definition and motivation
  11. Almost additivity of the Cartan projection
  12. A dichotomy formula
  13. Weak law of large numbers
  14. Central limit Theorem
  15. On the Generalized Central limit Theorem

Key Result

Theorem A

Let $\nu$ be a probability measure over $\mathrm{SL}(E)$ and let $(\gamma_n)_{n \ge 0} \sim \nu^{\otimes\mathbb{N}}$. Assume that $\Gamma_\nu$ is proximal, that $\int\kappa^2 d\nu = \mathbb{E}(\kappa(\gamma_0)^2) = + \infty$ and that $\kappa_* \nu$ is in the domain of attraction of a non-degenerate Then for all such sequences $(a_n)$ and $(b_n)$, there exists a constant $b \ge 0$ such that for al

Theorems & Definitions (62)

  • Theorem A: Convergence to stable law with infinite variance
  • Theorem B: Central limit Theorem with first moment assumption on the inverse
  • Definition 1.1
  • Proposition 1.2
  • proof
  • Theorem 1.3: Probabilistic bound on the norm-cancellation
  • Theorem 1.4: Pivoting technique
  • Theorem 1.5: Generalized Central-Limit-Theorem for the norm
  • Theorem 1.6: Weak law of large numbers with doubled exponent
  • Theorem 1.7
  • ...and 52 more