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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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