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 $\varphi\in L^{ 2}\left(m\right)$ 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(\left\lfloor \sqrt{ n}x\right\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.
Categories: math.DS