Note on limit distribution of normalized return times and escape rate

In this note we discuss limit distribution of normalized return times for shrinking targets and draw a necessary and sufficient condition using sweep-out sequence in order for the limit distribution to be exponential with parameter 1. The normalizing coefficients are the same as the sizes of the targets. Moreover we study escape rate, namely the exponential decay rate of sweep-out sequence and prove that in ψ-mixing systems for a certain class of sets the escape rate is in limit proportional to the size of the set.