The extremal index, hitting time statistics and periodicity

• Type: Article in International Scientific Journal
• Date: 2012

Authors

Ana Cristina M Moreira Freitas (Author) FEP; Jorge Milhazes Freitas (Author) FCUP; Mike Todd (Author) Other

Journal

Title: Advances in Mathematics

Vol. 231 No. 5

Pages: 2626-2665

ISSN: 0001-8708

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 59 Citations

ISI Web of Science

Scopus - 61 Citations

Scientific classification

FOS: Natural sciences > Mathematics

CORDIS: Physical sciences > Mathematics

Other information

Authenticus ID: P-002-377

DOI: 10.1016/j.aim.2012.07.029

Abstract (EN): The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. We derive new, easily checkable, conditions which identify Extreme Value Laws with particular extremal indices. In the dynamical context we prove that the extremal index is associated with periodic behaviour. The analogy of these laws in the context of hitting time statistics, as studied in the authors' previous works on this topic, is explained and exploited extensively allowing us to prove, for the first time, the existence of hitting time statistics for balls around periodic points. Moreover, for very well behaved systems (uniformly expanding) we completely characterise the extremal behaviour by proving that either we have an extremal index less than I at periodic points or equal to 1 at any other point. This theory then also applies directly to general stochastic processes, adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 40