Weak invariance principle for the local times of Gibbs-Markov processes

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 $\left(X,\mathcal{B},m,T,\alpha \right)$ is a mixing, probability preserving Gibbs-Markov{\normalsize{}. and let $\phi \in L^2\left(m\right)$ be an aperiodic function with mean $0$. Set $S_n={\sum }_{k=0}^n{X}_k$ and define the hitting time process $L_n\left(x\right)$ be the number of times $S_k$ hits $x\in \mathbb{Z}$ up to step $n.$ The normalized local time process $l_n\left(x\right)$ is defined by $l_n\left(t\right)=\frac{L_n\left(\lfloor \sqrt{n}x\rfloor \right)}{\sqrt{n}},x\in \mathbb{R}$. We prove under that $l_n\left(x\right)$ 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.