Hitting and return times in ergodic dynamical systems

Given an ergodic dynamical system $(X,T,\mu)$, and $U\subset X$ measurable with $\mu (U)>0$, let $\mu (U)\tau_U(x)$ denote the normalized hitting time of $x\in X$ to $U$. We prove that given a sequence $(U_n)$ with $\mu (U_n)\to 0$, the distribution function of the normalized hitting times to $U_n$ converges weakly to some sub-probability distribution $F$ if and only if the distribution function of the normalized return time converges weakly to some distribution function $\tilde F$, and that in the converging case, F(t)=\int_0^t(1-\tilde F(s))ds, t\ge 0.\tag$\diamondsuit$ This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is too.