[Paper Review] Central limit theorem for products of toral automorphisms

Jean-Pierre Conze, Stéphane Le Borgne+1 more|arXiv (Cornell University)|Jun 21, 2010

Mathematical Dynamics and Fractals14 references3 citations

TL;DR

This paper establishes a quenched central limit theorem (CLT) for sums of Hölder-continuous functions evaluated along random products of toral automorphisms defined by matrices in $SL(d,\mathbb{Z})$. Under proximality and total irreducibility conditions on the matrix set $\mathcal{A}$, it proves that for $\mathbb{P}$-almost every sequence of matrices, the normalized sum converges in distribution to a normal law with a positive variance, and provides a rate of convergence via the method of multiplicative systems and spectral gap techniques.

ABSTRACT

Let $(τ_n)$ be a sequence of toral automorphisms $τ_n : x ightarrow A_n x \hbox{mod}\ZZ^d$ with $A_n \in {\cal A}$, where ${\cal A}$ is a finite set of matrices in $SL(d, \mathbb{Z})$. Under some conditions the method of "multiplicative systems" of Komlòs can be used to prove a Central Limit Theorem for the sums $\sum_{k=1}^n f(τ_k \circ τ_{k-1} \cdots \circ τ_1 x)$ if $f$ is a Hölder function on $\mathbb{T}^d$. These conditions hold for $2 imes 2$ matrices with positive coefficients. In dimension $d$ they can be applied when $A_n= A_n(ω)$, with independent choices of $A_n(ω)$ in a finite set of matrices $\in SL(d, \mathbb{Z})$, in order to prove a "quenched" CLT.

Motivation & Objective

• To establish a quenched central limit theorem for dynamical systems driven by random compositions of toral automorphisms.
• To determine conditions under which the normalized sum of Hölder functions along random matrix products converges to a normal distribution almost surely in the matrix sequence.
• To quantify the rate of convergence to the normal distribution for such sums.
• To extend results on random walks on $SL(d,\mathbb{Z})$ to the context of non-i.i.d. dynamical systems on the torus.
• To analyze the variance of the normalized sum and prove its positivity almost surely for non-zero functions.

Proposed method

• Uses the method of multiplicative systems, originally developed by Komlòs, to control the characteristic function of the normalized sum.
• Applies spectral gap estimates for the random walk on $SL(d,\mathbb{Z})$ via results from Guivarc’h and Raugi to control the decay of correlations.
• Employs Fourier analysis on the torus, tracking the evolution of Fourier coefficients under the action of matrix products.
• Introduces a decomposition of the characteristic function into a product of terms corresponding to different frequency components, enabling separation of scales.
• Uses exponential moment bounds and estimates on the characteristic function $\mathbb{E}[e^{ix S_n / \|S_n\|_2}]$ to compare it with the Gaussian characteristic function $e^{-x^2/2}$.
• Applies concentration inequalities and truncation techniques to control the error in approximating the characteristic function by the Gaussian one, particularly under the condition $|x|\delta \leq 1$.

Experimental results

Research questions

• RQ1 Under what conditions does the normalized sum $\frac{1}{\sqrt{n}}\sum_{k=1}^n f(\tau_k \circ \cdots \circ \tau_1 x)$ converge in distribution to a normal law for $\mathbb{P}$-almost every sequence of matrices?
• RQ2 What is the rate of convergence of this normalized sum to the normal distribution?
• RQ3 How does the variance $\sigma(f)^2 = \lim_{n \to \infty} \frac{1}{n} \|S_n(f)\|_2^2$ behave for a non-zero Hölder function $f$?
• RQ4 Can the quenched CLT be established when the matrix sequence is random but the initial point $x$ is fixed?
• RQ5 What role do the spectral properties of the random walk on $SL(d,\mathbb{Z})$ play in ensuring the CLT?

Key findings

• For $\mathbb{P}$-almost every sequence $\omega$, the limit $\sigma(f) = \lim_{n \to \infty} \frac{1}{\sqrt{n}} \|S_n(f)\|_2$ exists and is positive for any non-zero Hölder function $f$ with zero mean.
• The normalized sum $\frac{1}{\sigma(f)\sqrt{n}} \sum_{k=1}^n f(\tau_k(\omega) \circ \cdots \circ \tau_1(\omega) \cdot)$ converges in distribution to $\mathcal{N}(0,1)$.
• The convergence to the normal distribution occurs with a rate of convergence that is quantified via bounds on the characteristic function difference $|\mathbb{E}[e^{ix S_n / \|S_n\|_2}] - e^{-x^2/2}|$.
• The proof relies on a spectral gap in $L^2_0$ for the random walk on $SL(d,\mathbb{Z})$, which ensures exponential decay of correlations.
• The result holds for both Hölder functions and characteristic functions of regular sets on the torus under the same conditions.
• The key technical innovation lies in the use of multiplicative systems and frequency separation to control the characteristic function, especially in the regime where $|x|\delta \leq 1$.