[Paper Review] Weak invariance principle for the local times of Gibbs-Markov processes

Michael Bromberg|arXiv (Cornell University)|Jun 16, 2014

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper establishes a functional weak invariance principle for the local times of Gibbs-Markov processes, proving that the normalized local time process $ l_n(x) $, derived from a mixing, probability-preserving Gibbs-Markov map with mean-zero, aperiodic, Lipschitz-continuous observables, converges in distribution to the local time of standard Brownian motion. The result extends to subshifts of finite type with Gibbs measures.

ABSTRACT

The subject of this paper is to prove a functional weak invariance principle for the local time of a process generated by a Gibbs-Markov map. More precisely, let $\left(X,\mathcal{B},m,T,α ight)$ is a mixing, probability preserving Gibbs-Markov{ ormalsize{}. and let $φ\in L^{2}\left(m ight)$ be an aperiodic function with mean $0$. Set $S_{n}=\sum_{k=0}^{n}X_{k}$ and define the hitting time process $L_{n}\left(x ight)$ be the number of times $S_{k}$ hits $x\in\mathbb {Z}$ up to step $n.$ The normalized local time process $l_{n}\left(x ight)$ is defined by $ l_{n}\left(t ight)=\frac{L_{n}\left(\left\lfloor \sqrt{n}x ight floor ight)}{\sqrt{n}},\,\, x\in\mathbb{R}$. We prove under that $l_{n}\left(x ight)$ converges in distribution to the local time of the Brownian Motion. The proof also applies to the more classical setting of local times derived from a subshift of finite type endowed with a Gibbs measure.

Motivation & Objective

• To establish a functional weak invariance principle for the local time of a Gibbs-Markov process.
• To prove convergence in distribution of the normalized local time process $ l_n(x) $ to the local time of Brownian motion.
• To extend the functional central limit theorem to the local time level for dynamical systems with Gibbs-Markov structure.
• To verify tightness and identify the unique limit of subsequences of $ l_n $, confirming convergence to the Brownian local time.

Proposed method

• Define the normalized local time process $ l_n(x) = \frac{\#\{0 \leq k \leq n : S_k = \lfloor \sqrt{n}x \rfloor\}}{\sqrt{n}} $, where $ S_k $ is the partial sum of an observable $ \varphi \in L^2(m) $.
• Use the transfer operator $ \hat{T} $ and its quasi-compactness on $ L_{\infty,\beta} $ to analyze the spectral properties of the system.
• Establish tightness of $ l_n $ in the Skorokhod space of cadlag functions on $ \mathbb{R} $ via moment bounds and Hölder continuity estimates.
• Prove convergence of occupation measures via approximation of $ \nu_{\omega_n} $ by $ \int l_n(x)dx $, showing $ \nu_{\omega_n} - \int l_n(x)dx \overset{d}{\to} 0 $.
• Identify the limit of any convergent subsequence of $ l_n $ as the local time of Brownian motion by showing convergence of projections $ \pi_g(l_n) \to \pi_g(l) $ for all $ g \in \bigcup_k G_k $.
• Leverage the continuity of the occupation time functional on the Skorokhod topology and the functional central limit theorem for $ \omega_n \to \omega $.

Experimental results

Research questions

• RQ1 Does the normalized local time process of a Gibbs-Markov system converge in distribution to the local time of Brownian motion?
• RQ2 Under what conditions on the observable $ \varphi $ does the functional weak invariance principle hold at the level of local times?
• RQ3 Can tightness of the local time process $ l_n $ be established under Gibbs-Markov dynamics with aperiodic, mean-zero, Lipschitz-continuous observables?
• RQ4 Is the local time of Brownian motion the unique weak limit of $ l_n $, and how can this be identified via projections on intervals?
• RQ5 To what extent does the result extend to subshifts of finite type equipped with Gibbs measures?

Key findings

• The normalized local time process $ l_n(x) $ converges in distribution to the local time $ l(x) $ of standard Brownian motion under the stated conditions.
• Tightness of $ l_n $ is established via moment bounds and Hölder continuity estimates, ensuring relative compactness in the space of cadlag functions.
• The difference between the occupation measure $ \nu_{\omega_n} $ and the integral of $ l_n $ converges in distribution to zero, confirming $ l_n $ as a density approximation.
• The limit of any convergent subsequence of $ l_n $ is almost surely equal to the local time of Brownian motion, as shown by convergence of projections on intervals.
• The proof relies on the functional central limit theorem for $ \omega_n \to \omega $, and the continuity of the occupation time functional on the Skorokhod topology.
• The result holds for subshifts of finite type with Gibbs measures, extending the applicability beyond general Gibbs-Markov maps.