Applications of weak dependence to extremes value theory and resampling

Résumé

The talk initially introduces the notion of weak dependence (Doukhan & Louhichi (1999) which is an extension to Rosenblatt 1956's strong mixing definition. Mixing notions were extremely successful (see Doukhan 1994), anyway they do not cover all the reasonable time series to be considered in statistics. Doukhan & Louhichi (1999)'s weak dependence is also adapted to resampling as noticed by Bickel and Buhlmann (1999). The notion of weak dependence allows development of limit theory with numerous applications covered by the textbook of Dedecker et al. 2007. The present talk considers results related to Extremes Value Theory (EVT) in the dependent setting. With Adam Jakubowski we develop an EVT avoiding for example ergodicity conditions of Markov processes. Based on O'Brian work we consider simple conditions for the existence of phantoms. This means that EVT of a stationary times series is the same as for some iid sample, in case some simple mixing type condition is asserted. Anyway this is not always easy to recover the tails of extremes from their theoretical expressions. Resampling is an essential tool to achieve it as this was proved in a discussion paper with Christian Robert and Silika Prohl (2011).