Abstract:
In this talk, we start by presenting some notions of extreme value theory, namely the notions of extreme value laws and extremal index, which is a parameter that quantifies the intensity of clustering of exceedances. Then, we consider discrete time dynamical systems and stochastic processes arising by evaluating an observable function along the orbits of such systems. We show the connection between extreme value laws and hitting time statistics and exploit it both in the absence and presence of clustering. We associate the occurrence of periodic phenomena to the existence of an extremal index less than 1.

Abstract:
We consider time series corresponding to the evaluation of given potential along the orbits of systems with `nice' first return induced maps. Nice, here, means good memory loss properties. We are particularly interested in events corresponding to exceedances of high thresholds, i.e. instants at which the potential reaches abnormally high values, and, more specifically, when these exceedances have a tendency to appear in clusters. We will see the consequences of clustering in the convergence of conveniently normalised point processes that count the number of exceedances in a certain time frame.