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Example of a non-standard extreme-value law

Published online by Cambridge University Press:  04 June 2014

NICOLAI HAYDN  and
MICHAL KUPSA
NICOLAI HAYDN
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, 90089-2532, USA email nhaydn@usc.edu
MICHAL KUPSA
Affiliation:
Institute of Information Theory and Automation, The Academy of Sciences of the Czech Republic, Prague 8, CZ-18208, Czech Republic email kupsa@utia.cas.cz Faculty of Information Technology, Czech Technical University in Prague, Prague 6, CZ-16000, Czech Republic
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Abstract

It has been shown that sufficiently well mixing dynamical systems with positive entropy have extreme-value laws which in the limit converge to one of the three standard distributions known for independently and identically distributed processes, namely Gumbel, Fréchet and Weibull distributions. In this short note, we give an example which has a non-standard limiting distribution for its extreme values. Rotations of the circle by irrational numbers are used and it will be shown that the limiting distribution is a step function where the limit has to be taken along a suitable sequence given by the convergents.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 6 , September 2015 , pp. 1902 - 1912
Copyright
© Cambridge University Press, 2014 

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References

Abadi, M.. Hitting, returning and the short correlation function. Bull. Braz. Math. Soc. (N.S.) 37(4) (2006), 1–17.Google Scholar
Collet, P.. Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 21 (2001), 401–420.CrossRefGoogle Scholar
Coelho, Z. and de Faria, E.. Limit laws of entrance times for homeomorphisms of the circle. Israel J. Math. 93 (1996), 93–112.CrossRefGoogle Scholar
Freitas, A. C. M. and Freitas, J. M.. On the link between dependence and independence in Extreme Value Theory for Dynamical Systems. Statist. Probab. Lett. 78 (2008), 1088–1093.CrossRefGoogle Scholar
Freitas, A. C. M. and Freitas, J. M.. Extreme values for Benedicks Carleson maps. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1117–1133.CrossRefGoogle Scholar
Freitas, A. C. M., Freitas, J. M. and Todd, M.. Hitting time statistics and extreme value theory. Probab. Theory Related Fields 147(3) (2010), 675–710.CrossRefGoogle Scholar
Freitas, A. C. M., Freitas, J. M. and Todd, M.. Extreme value laws in dynamical systems for non-smooth observations. J. Stat. Phys. 142(1) (2011), 108–126.CrossRefGoogle Scholar
Gupta, C., Holland, M. and Nicol, M.. Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. Ergod. Th. & Dynam. Sys. 31(5) (2011), 1363–1390.CrossRefGoogle Scholar
Haydn, N.. Entry and return times distribution. Dyn. Syst. 28(3) (2013), 333–353.CrossRefGoogle Scholar
Holland, M., Nicol, M. and Török, A.. Extreme value distributions for non-uniformly hyperbolic dynamical systems. Trans. Amer. Math. Soc. 364(2) (2012), 661–688.CrossRefGoogle Scholar
Kim, D. H. and Park, K. K.. The first return time properties of an irrational rotation. Proc. Amer. Math. Soc. 136(11) (2008), 3941–3951.CrossRefGoogle Scholar
Kupsa, M. and Lacroix, Y.. Asymptotics for hitting times. Ann. Probab. 33(2) (2005), 610–619.CrossRefGoogle Scholar