[Paper Review] Empirical central limit theorems for ergodic automorphisms of the torus

Jérôme Dedecker, Florence Merlevède+1 more|arXiv (Cornell University)|Oct 12, 2012

Mathematical Dynamics and Fractals17 references22 citations

TL;DR

This paper establishes weak convergence of the sequential empirical process for ergodic automorphisms of the torus under mild continuity conditions on the observable function. Using new moment inequalities for non-adapted sequences and conditional expectation estimates, it proves convergence to a Gaussian process in both $L^p$ and uniform topology, extending functional central limit theorems to dependent, non-i.i.d. dynamical systems on the torus.

ABSTRACT

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f o T^i)_{i \geq 1} under some mild conditions on the modulus of continuity of f. The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural filtration.

Motivation & Objective

• To establish weak convergence of the sequential empirical process for ergodic automorphisms of the $d$-dimensional torus.
• To extend functional central limit theorems to dependent, non-i.i.d. sequences arising from deterministic dynamical systems.
• To provide conditions on the modulus of continuity of observables ensuring weak convergence of empirical processes to Gaussian limits.
• To develop new probabilistic tools—moment inequalities and conditional expectation estimates—for non-adapted, weakly dependent sequences in dynamical systems.
• To enable applications to partial sums, including weak invariance principles and rates of convergence in strong invariance principles.

Proposed method

• Derive new central limit theorems for dependent sequences in smooth Banach spaces, particularly $L^p$-valued processes.
• Establish a new Rosenthal-type inequality for non-adapted, weakly dependent sequences, crucial for controlling moments of empirical processes.
• Introduce a key estimate (Theorem 19) controlling conditional expectations of observables with respect to a filtration from Lind’s partition, using the modulus of continuity of $f$.
• Use the spectral properties of ergodic toral automorphisms and the structure of the associated linear map $S$ to control decay of dependence.
• Apply the theory of Kiefer processes to characterize the limiting Gaussian process in the uniform topology on $[0,1] \times \mathbb{R}^\ell$.
• Leverage the connection between the modulus of continuity of $f$ and the decay of conditional expectations to derive integrability conditions on Fourier coefficients or log-singularities.

Experimental results

Research questions

• RQ1 Under what conditions on the modulus of continuity of $f$ does the sequential empirical process $\{n^{-1/2}S_{[nt]}(s)\}$ converge weakly to a Gaussian process?
• RQ2 Can moment inequalities for non-adapted sequences be extended to dynamical systems with long-range dependence, such as non-hyperbolic toral automorphisms?
• RQ3 How can conditional expectation estimates be controlled using only the modulus of continuity of $f$, without requiring Hölder regularity?
• RQ4 What are the implications of these results for the weak invariance principle and rates of convergence in the strong invariance principle for partial sums?
• RQ5 To what extent can the limiting behavior of empirical processes on the torus be characterized using Kiefer process-type limits in the dependent case?

Key findings

• The sequential empirical process $\{n^{-1/2}S_{[nt]}(s), t \in [0,1], s \in \mathbb{R}^\ell\}$ converges weakly in $\ell^\infty([0,1] \times \mathbb{R}^\ell)$ to a generalized Kiefer process, a Gaussian process with independent increments in time and a covariance structure depending on $f$.
• For $\ell=1$, the process $\{n^{-1/2}S_{[nt]}(s)\}$ converges weakly in $D_{L^p}([0,1])$ to a $L^p$-valued Wiener process, with a well-characterized covariance operator.
• The condition $\sum_{n>0} \frac{\omega(f, \zeta^n)}{\sqrt{n}} < \infty$ implies $\left\| \max_{1 \leq k \leq n} \left| \sum_{i=1}^k (f \circ T^i - \bar{\lambda}(f)) \right| \right\|_p \ll n^{1/2}$ for any $p > 2$, ensuring moment bounds.
• The integral condition $\int_0^{1/2} \frac{\omega(f,t)}{t |\log t|^{1/2}} dt < \infty$ ensures the absolute convergence of the variance series $\sigma^2(f)$ and the weak invariance principle for partial sums.
• For $a > 1/2$, the rate of convergence in the strong invariance principle is $o(n^{1/2 - \epsilon})$ almost surely; for $a \geq 3/2$, the rate improves to $n^{1/4} \log n$.
• The key estimate (Theorem 19) provides a control of conditional expectations $\mathbb{E}_0[g^{(0)}_{s,t}(X_j) g^{(0)}_{s,t}(X_{j+i})]$ in terms of $\omega(f, \cdot)$, enabling tightness and convergence proofs without requiring Hölder regularity.