  We study convergence of return- and hitting-time distributions of small sets
$E_{k}$ with $\mu(E_{k})\rightarrow0$ in recurrent ergodic dynamical systems
preserving an infinite measure $\mu$. Some properties which are easy in finite
measure situations break down in this null-recurrent setup. However, in the
presence of a uniform set $Y$ with wandering rate regularly varying of index
$1-\alpha$ with $\alpha\in(0,1]$, there is a scaling function suitable for all
subsets of $Y$. In this case, we show that return distributions for the $E_{k}$
converge iff the corresponding hitting time distributions do, and we derive an
explicit relation between the two limit laws. Some consequences of this result
are discussed. In particular, this leads to improved sufficient conditions for
convergence to $\mathcal{E}^{1/\alpha}\,\mathcal{G}_{\alpha}$, where
$\mathcal{E}$ and $\mathcal{G}_{\alpha}$ are independent random variables, with
$\mathcal{E}$ exponentially distributed and $\mathcal{G}% _{\alpha}$ following
the one-sided stable law of order $\alpha$ (and $\mathcal{G}_{1}:=1$). The same
principle also reveals the limit laws (different from the above) which occur at
hyperblic periodic points of prototypical null-recurrent interval maps. We also
derive similar results for the barely recurrent $\alpha=0$ case.
