  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 $X_n$.
