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Edgeworth expansion and large deviations for the coefficients of products of positive random matrices

Hui Xiao, Ion Grama, Quansheng Liu

TL;DR

This work analyzes products $G_n=g_n\cdots g_1$ of i.i.d. positive $d\times d$ matrices, focusing on the entries $G_n^{i,j}$ and coefficients $\langle f, G_n v\rangle$. Under FK-weak and non-lattice conditions, it establishes Berry-Esseen bounds and a first-order Edgeworth expansion under the optimal third moment, and derives exact upper and lower large deviation asymptotics via a change-of-measure framework, with uniform control across parameters. The authors develop a comprehensive spectral gap theory for the norm cocycle and the coefficient cocycle, including perturbations $R_{it}$, $P_s$, $R_{s,z}$, and their variants, to underpin the limit theorems and large deviations. These techniques yield local limit theorems with large deviations for coefficients, LLTs for the matrix norm and spectral radius, and bounds for $\rho(G_n)$, providing a unified probabilistic-operator approach to products of positive random matrices with minimal moment assumptions. The results have implications for related stochastic models (e.g., branching processes in random environments and perpetuity-type sequences) and introduce a novel cocycle framework $\sigma_f(g,v)=\log\frac{\langle f,gv\rangle}{\langle f,v\rangle}$ with independent interest.

Abstract

Consider the matrix products $G_n: = g_n \ldots g_1$, where $(g_{n})_{n\geq 1}$ is a sequence of independent and identically distributed positive random $d\times d$ matrices. Under the optimal third moment condition, we first establish a Berry-Esseen theorem and an Edgeworth expansion for the $(i,j)$-th entry $G_n^{i,j}$ of the matrix $G_n$, where $1 \leq i, j \leq d$. Using the Edgeworth expansion for $G_n^{i,j}$ under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries $G_n^{i,j}$ subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for $G_n^{i,j}$ and upper and lower large deviations bounds for the spectral radius $ρ(G_n)$ of $G_n$. A byproduct of our approach is the local limit theorem for $G_n^{i,j}$ under the optimal second moment condition. In the proofs we develop a spectral gap theory for the norm cocycle and for the coefficients, which is of independent interest.

Edgeworth expansion and large deviations for the coefficients of products of positive random matrices

TL;DR

This work analyzes products of i.i.d. positive matrices, focusing on the entries and coefficients . Under FK-weak and non-lattice conditions, it establishes Berry-Esseen bounds and a first-order Edgeworth expansion under the optimal third moment, and derives exact upper and lower large deviation asymptotics via a change-of-measure framework, with uniform control across parameters. The authors develop a comprehensive spectral gap theory for the norm cocycle and the coefficient cocycle, including perturbations , , , and their variants, to underpin the limit theorems and large deviations. These techniques yield local limit theorems with large deviations for coefficients, LLTs for the matrix norm and spectral radius, and bounds for , providing a unified probabilistic-operator approach to products of positive random matrices with minimal moment assumptions. The results have implications for related stochastic models (e.g., branching processes in random environments and perpetuity-type sequences) and introduce a novel cocycle framework with independent interest.

Abstract

Consider the matrix products , where is a sequence of independent and identically distributed positive random matrices. Under the optimal third moment condition, we first establish a Berry-Esseen theorem and an Edgeworth expansion for the -th entry of the matrix , where . Using the Edgeworth expansion for under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for and upper and lower large deviations bounds for the spectral radius of . A byproduct of our approach is the local limit theorem for under the optimal second moment condition. In the proofs we develop a spectral gap theory for the norm cocycle and for the coefficients, which is of independent interest.
Paper Structure (21 sections, 32 theorems, 266 equations)

This paper contains 21 sections, 32 theorems, 266 equations.

Table of Contents

  1. Introduction and main results
  2. Background and objectives
  3. Edgeworth expansion and Berry-Esseen theorem
  4. Upper and lower large deviations
  5. Applications
  6. Spectral gap theory for the norm cocycle
  7. Preliminary statements
  8. Spectral gap properties of the perturbed operator $R_{it}$
  9. Spectral gap properties of the transfer operator $P_s$
  10. Spectral gap properties of the perturbed transfer operator $R_{s, z}$
  11. Spectral gap theory for the coefficients
  12. Spectral gap properties of the perturbed operator $R_{it, f}$
  13. Spectral gap properties of the transfer operator $P_{s,f}$
  14. Spectral gap properties of the perturbed transfer operator $R_{s,z,f}$
  15. Proof of Edgeworth expansion and Berry-Esseen theorem
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume Condi-FK-weak, Condi-NonLattice and $\int_{\mathbb{G}} \log^3 N(g) \mu(dg) < \infty$. Then, for any compact set $K \subset \mathbb{R}^{d}_+ \setminus \{0\}$, we have, as $n \to \infty$, uniformly in $f, v \in K$ and $y \in \mathbb{R}$,

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: FK60
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 44 more