Extreme Value Theory techniques have frequently been used in the dynamical contest, starting with Collet's 2001 paper, to study the recurrence of points in the space to a specific point or set. This has since been extended to prove Poisson limit laws, which require some asymptotic independence in the system as well as local recurrence properties around the point/set. Collet also considered self-recurrence of points, for which there are more dependence properties to overcome, and proved an average recurrence result. I will describe work with Mark Holland where we extend this to get Poisson limit laws, with some averaged parameters. I will also outline some almost-sure results and connections to recurrence more generally.