On distributional limit laws for recurrence

For a probability measure preserving dynamical system $(\mathcal{X},f,\mu)$, the Poincar\'e Recurrence Theorem asserts that $\mu$-almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process $X_n(x)=\text{dist}(f^n(x),x))$, and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-$n$ counting process $R_n(x)$ associated to the number recurrences below a certain radii sequence $r_n(\tau)$ follows an \emph{averaged} Poisson distribution $G(\tau)$. Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process $X_n$.