[Paper Review] Central Limit Theorems for Non-Invertible Measure Preserving Maps

Michael C. Mackey, Marta Tyran‐Kamińska|ArXiv.org|Aug 25, 2006

Mathematical Dynamics and Fractals25 references3 citations

TL;DR

This paper establishes a functional central limit theorem for non-invertible measure-preserving maps without requiring ergodicity, using the Perron-Frobenius operator to analyze the convergence of Birkhoff sums to a scaled Brownian motion. The key contribution is a new sufficient condition involving the $ L^2 $-norm decay of iterated transfer operators, which generalizes previous results to non-ergodic settings and provides a series expansion for the limiting variance.

ABSTRACT

We establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator. We apply the result to asymptotically periodic transformations and give an extensive specific example of asymptotically periodic transformations by using the tent map.

Motivation & Objective

• To extend functional central limit theorems to non-invertible, non-ergodic measure-preserving transformations.
• To establish a sufficient condition for weak convergence of Birkhoff sums to a scaled Brownian motion in the Skorohod space.
• To provide a series expansion for the limiting variance $ \eta = \mathbb{E}_\nu(\tilde{h}^2 \mid \mathcal{I}) $ in terms of the transfer operator and iterates of the map.
• To apply the result to asymptotically periodic systems, including the tent map, as a concrete example.

Proposed method

• The analysis uses the Perron-Frobenius operator $ \mathcal{P}_T $ to study the $ L^2 $-norm decay of the sum $ \sum_{k=0}^{n-1} \mathcal{P}_T^k h $.
• A new sufficient condition is introduced: $ \sum_{n=1}^\infty n^{-3/2} \left\| \sum_{k=0}^{n-1} \mathcal{P}_T^k h \right\|_2 < \infty $, ensuring weak convergence to $ \sqrt{\eta} w $.
• The limiting variance $ \eta $ is characterized as $ \mathbb{E}_\nu(\tilde{h}^2 \mid \mathcal{I}) $, where $ \tilde{h} $ satisfies $ \mathcal{P}_T \tilde{h} = 0 $ and $ \left\| \frac{1}{\sqrt{n}} \sum_{j=0}^{n-1} (h - \tilde{h}) \circ T^j \right\|_2 \to 0 $.
• A series expansion for $ \eta $ is derived via dyadic decomposition: $ \eta = \mathbb{E}_\nu(h^2 \mid \mathcal{I}) + \sum_{j=0}^\infty \frac{\mathbb{E}_\nu(S_{2^j} S_{2^j} \circ T^{2^j} \mid \mathcal{I})}{2^j} $.
• The proof relies on maximal inequalities and $ L^2 $-boundedness techniques inspired by Peligrad and Utev, with conditional expectation and contraction properties used to control the series convergence.

Experimental results

Research questions

• RQ1 Under what conditions does the Birkhoff sum of a function under a non-invertible, non-ergodic measure-preserving map converge weakly to a scaled Brownian motion?
• RQ2 How can the limiting variance $ \eta $ be expressed explicitly in terms of the transfer operator and invariant $ \sigma $-algebra $ \mathcal{I} $?
• RQ3 Can the functional central limit theorem be extended beyond the ergodic case, particularly to asymptotically periodic systems?
• RQ4 What is the role of the Perron-Frobenius operator in characterizing the convergence rate and variance in non-ergodic settings?

Key findings

• The functional central limit theorem holds under the condition $ \sum_{n=1}^\infty n^{-3/2} \left\| \sum_{k=0}^{n-1} \mathcal{P}_T^k h \right\|_2 < \infty $, which generalizes earlier results to non-ergodic maps.
• The limiting process is $ \sqrt{\eta} w $, where $ \eta = \mathbb{E}_\nu(\tilde{h}^2 \mid \mathcal{I}) $, and $ \tilde{h} $ is a function in $ L^2 $ with $ \mathcal{P}_T \tilde{h} = 0 $ such that the difference $ h - \tilde{h} $ becomes negligible in $ L^2 $-norm after averaging.
• For ergodic maps, $ \eta $ reduces to a constant $ \|h\|_2^2 $, recovering the classical functional central limit theorem.
• A series expansion for $ \eta $ is derived: $ \eta = \mathbb{E}_\nu(h^2 \mid \mathcal{I}) + \sum_{j=0}^\infty \frac{\mathbb{E}_\nu(S_{2^j} S_{2^j} \circ T^{2^j} \mid \mathcal{I})}{2^j} $, convergent under the condition $ \sum_{j=0}^\infty 2^{-j/2} \left\| \sum_{k=1}^{2^j} \mathcal{P}_T^k h \right\|_2 < \infty $.
• The result applies to asymptotically periodic transformations, and the tent map is used as a detailed example where the conditions are verified.