Hitting and return times in ergodic dynamical systems

Given an ergodic dynamical system (X, T, mu), and U subset of X measurable with mu(U) > 0, let mu(U)tau(U)(x) denote the normalized hitting time of x is an element of X to U. We prove that given a sequence (U-n) with mu(U-n) -> 0, the distribution function of the normalized hitting times to U-n converges weakly to some subprobability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution function and that in the converging case, [GRAPHICS] 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 also.