Laws of rare events with convergence rates.



Jorge Milhazes de Freitas,

Faculdade de Ciências - Universidade do Porto/ CMUP



Abstract: In the classical theory of Extreme Values, it is well known that the Extremal Index (EI) measures the intensity of clustering of extreme events in stationary processes. By extreme or rare events, we mean occurrences usually undesired and possibly catastrophic that have a small probability of occurring. We will see that for some certain uniformly expanding systems there exists a dichotomy based on whether the rare events correspond to the entrance in small balls around a periodic point or a non-periodic point. In fact, either there exists EI in (0,1) around (repelling) periodic points or the EI is equal to 1 at every non-periodic point. The main assumption is that the systems have sufficient decay of correlations of observables in some Banach space against all integrable observables. Under the same assumption, we obtain convergence rates for the asymptotic extreme value limit distribution. The dependence of the error terms on the `time' and `length' scales is made very explicit.





Location: C6.2.38, Faculdade de Ciências, UL

Supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2013