[Paper Review] Stein's method for dynamical systems

Olli Hella, Juho Leppänen+1 more|arXiv (Cornell University)|Jan 11, 2017

Mathematical Dynamics and Fractals43 references3 citations

TL;DR

This paper adapts Stein's method to dynamical systems, providing a multivariate central limit theorem with a computable rate of convergence for both discrete- and continuous-time systems. By leveraging correlation decay and avoiding characteristic functions, it establishes Berry–Esséen-type bounds using a direct approach that yields convergence rates with minimal additional work beyond proving the central limit theorem.

ABSTRACT

We present an adaptation of Stein's method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method, which yields a convergence rate essentially with the same amount of work as the central limit theorem, together with a multiplicative constant that can be computed directly from the assumptions.

Motivation & Objective

• To extend Stein’s method of normal approximation to discrete- and continuous-time dynamical systems.
• To derive a multivariate central limit theorem with a computable rate of convergence for observables of measure-preserving systems.
• To provide a systematic scheme for verifying correlation decay conditions in concrete dynamical systems.
• To demonstrate that the method yields convergence rates with the same effort as proving the central limit theorem alone.
• To enable application to non-stationary settings and quasistatic systems, extending beyond traditional i.i.d. or stationary assumptions.

Proposed method

• Adapts Stein’s method to dynamical systems by replacing characteristic function techniques with correlation decay analysis.
• Uses a coupling-based approach through auxiliary randomization, but circumvents it via structural rigidity in hyperbolic systems.
• Employs a decomposition of the test function and integration over parameterized families of observables to control error terms.
• Applies Markov’s inequality and $L^1$-bounds on return times to handle small denominators in the coupling construction.
• Derives explicit bounds using decay rates $\theta^k$ for correlations, with $\theta < 1$, derived from mixing properties.
• Introduces a parameterized family of measures $\nu_q$ and a partition $\mathcal{Q}$ to localize and control error contributions.

Experimental results

Research questions

• RQ1 Can Stein’s method be adapted to provide rates of convergence in the multivariate central limit theorem for dynamical systems?
• RQ2 How can correlation decay conditions be formulated and verified to ensure normal approximation with explicit error bounds?
• RQ3 What is the minimal amount of work required to obtain a rate of convergence using Stein’s method compared to classical approaches?
• RQ4 Can the method be extended to non-stationary or quasistatic dynamical systems where a common invariant measure does not exist?
• RQ5 How does the method compare to Rio’s or Pène’s approaches in terms of flexibility and applicability to higher-dimensional systems?

Key findings

• The method yields a Berry–Esséen-type bound for the multivariate central limit theorem in dynamical systems with a rate of convergence of order $O(\theta^{K/4})$, where $\theta < 1$ is a decay rate derived from mixing properties.
• The convergence rate is obtained with the same level of effort as proving the central limit theorem, with a multiplicative constant computable directly from the assumptions.
• The bound depends explicitly on the dimension $d$, the $L^\infty$-norm of the observable $f$, the Lipschitz norm of the test function $B$, and the decay parameters $\vartheta, \vartheta_0, \vartheta_1$.
• The method avoids characteristic functions and instead relies on correlation decay, enabling application to deterministic systems with strong mixing behavior.
• A key technical innovation is the use of a parameterized family of measures and a localization argument to control singularities from small return times.
• The final bound is of the form $\left| \int f^n \cdot B(W^n) \, d\mu \right| \leq M \left( dH\|B\|_{\infty} + d\|f\|_{\infty}\|B\|_{\infty} + \frac{d^2\|f\|_{\infty}\|DB\|_{\infty}}{\sqrt{N}} \frac{H}{1-\vartheta} \right) \theta^{K/4} $, with $\theta = \max(\vartheta, \vartheta_0^2, \vartheta_1^4, e^{-1/c_0})$.