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Central Limit Theorem for non-stationary random products of $\SL(2, \R)$ matrices

Anton Gorodetski, Victor Kleptsyn, Grigorii Monakov

TL;DR

This work proves a Central Limit Theorem for non-stationary products of ${SL}(2,\mathbb{R})$ matrices under a compactness and moment condition: if ${\mathcal K}$ is a compact measure set satisfying a no-deterministic-image condition and ${\mathbb E}_\mu (\log \|A\|)^\gamma$ is finite with $\gamma>2$, then the normalized fluctuation of the log-norm, $$\frac{\log \|T_n\| - L_n}{\sqrt{\mathrm{Var}(\log \|T_n\|)}},$$ converges in distribution to a standard normal, uniformly over choices of $(\mu_1,\mu_2,\dots)\in {\mathcal K}$. The proof builds a non-stationary analog of the Le Page–Tutubalin program by introducing a discrepancy term $R_{n,n'}$, establishing uniform moment bounds, proving linear variance growth, and applying a bootstrapping argument to transfer independence into Gaussian convergence. The results extend to the distribution of logs of vector images and matrix elements, and the paper also demonstrates that the moment condition $\gamma>2$ is optimal by providing non-Gaussian limits under weaker assumptions. The methodology combines non-stationary log-Hölder estimates on the projective line, careful moment control, and a robust bootstrapping framework for distance to Gaussian.

Abstract

We prove Central Limit Theorem for non-stationary random products of $SL(2, \mathbb{R})$ matrices, generalizing the classical results by Le Page and Tutubalin that were obtained in the case of iid random matrix products.

Central Limit Theorem for non-stationary random products of $\SL(2, \R)$ matrices

TL;DR

This work proves a Central Limit Theorem for non-stationary products of matrices under a compactness and moment condition: if is a compact measure set satisfying a no-deterministic-image condition and is finite with , then the normalized fluctuation of the log-norm, converges in distribution to a standard normal, uniformly over choices of . The proof builds a non-stationary analog of the Le Page–Tutubalin program by introducing a discrepancy term , establishing uniform moment bounds, proving linear variance growth, and applying a bootstrapping argument to transfer independence into Gaussian convergence. The results extend to the distribution of logs of vector images and matrix elements, and the paper also demonstrates that the moment condition is optimal by providing non-Gaussian limits under weaker assumptions. The methodology combines non-stationary log-Hölder estimates on the projective line, careful moment control, and a robust bootstrapping framework for distance to Gaussian.

Abstract

We prove Central Limit Theorem for non-stationary random products of matrices, generalizing the classical results by Le Page and Tutubalin that were obtained in the case of iid random matrix products.

Paper Structure

This paper contains 20 sections, 33 theorems, 229 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Preliminaries and main results
  4. Notations and plan of the proof of the main result
  5. Moment estimates for $R_{n,n'}$
  6. Preliminaries: action of ${\mathrm{SL}}(2,{\mathbb R})$ matrices
  7. Estimates
  8. Moments growth for $\xi_n$
  9. Probabilistic setup
  10. Upper bounds on moments of $\widetilde{\xi}_{a;n}$
  11. Linear growth of variances
  12. Bootstrapping: distance to the Gaussian distribution
  13. Preliminaries
  14. Initial estimates
  15. Sum of two independent variables
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\{X_k, k\ge 1\}$ be independent and identically distributed random variables, taking values in ${\mathrm{SL}}(d, \mathbb{R})$, the $d\times d$ matrices with determinant one, let $G_X$ be the smallest closed subgroup of ${\mathrm{SL}}(d, \mathbb{R})$ containing the support of the distribution of Also, assume that $G_X$ is not compact and is strongly irreducible, i.e. there exists no $G_X$-inva

Figures (4)

  • Figure 1: Left: mostly contracted direction $\mathbf{r}(B)$ and a vector $v$. Center: their images after rotation by ${\mathrm{Rot}}_{\beta_2}$. Right: the images after the application of the diagonal matrix in the decomposition \ref{['eq:B-diag']}.
  • Figure 2: Top left: image $[B_1 e_i]$ provided by Corollary \ref{['cor:log-2']}. Top right: the direction $\mathbf{r}(B_2)$ and the preimage $[B_2^{-1} e_j]$, sufficiently close to it, that is provided by Lemma \ref{['l:B-preimage']}. Bottom: these three directions and an arc of length at least $\Delta_{B_1,B_2}$.
  • Figure 3: The set $L$ and its preimages
  • Figure 4: Density for the limit law in Example \ref{['ex:no-CLT-1D']}

Theorems & Definitions (71)

  • Theorem 1.1: H. Furstenberg Fur1
  • Remark 1.2
  • Theorem 1.3: Benuist, Quint, BQ1
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 61 more