Spontaneous clustering in theoretical and some empirical stationary processes*

T. Downarowicz; Y. Lacroix; D. Léandri

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 256-262
  • ISSN: 1292-8100

Abstract

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In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for “rare events” defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation.

How to cite

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Downarowicz, T., Lacroix, Y., and Léandri, D.. "Spontaneous clustering in theoretical and some empirical stationary processes*." ESAIM: Probability and Statistics 14 (2010): 256-262. <http://eudml.org/doc/250850>.

@article{Downarowicz2010,
abstract = { In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for “rare events” defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation. },
author = {Downarowicz, T., Lacroix, Y., Léandri, D.},
journal = {ESAIM: Probability and Statistics},
keywords = {Stationary random process; return time; hitting time; attracting; limit law; cluster; the law of series; stationary random process},
language = {eng},
month = {10},
pages = {256-262},
publisher = {EDP Sciences},
title = {Spontaneous clustering in theoretical and some empirical stationary processes*},
url = {http://eudml.org/doc/250850},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Downarowicz, T.
AU - Lacroix, Y.
AU - Léandri, D.
TI - Spontaneous clustering in theoretical and some empirical stationary processes*
JO - ESAIM: Probability and Statistics
DA - 2010/10//
PB - EDP Sciences
VL - 14
SP - 256
EP - 262
AB - In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for “rare events” defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation.
LA - eng
KW - Stationary random process; return time; hitting time; attracting; limit law; cluster; the law of series; stationary random process
UR - http://eudml.org/doc/250850
ER -

References

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  4. T. Downarowicz and Y. Lacroix, The Law of Series, .  Zbl1220.37008URIhttp://arXiv.org/abs/math/0601166
  5. N. Haydn, Y. Lacroix and S. Vaienti, Entry and return times in ergodic aperiodic dynamical systems. Ann. Probab.33 (2005) 2043–2050.  Zbl1130.37305
  6. M. Kac, On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc.53 (1947) 1002–1010.  Zbl0032.41802
  7. M. Kupsa and Y. Lacroix, Asymptotics for hitting times. Ann. Probab.33 (2005) 610–619.  Zbl1065.37006
  8. Y. Lacroix, Possible limit laws for entrance times of an ergodic aperiodic dynamical system. Israel J. Math.132 (2002) 253–263.  Zbl1054.37001
  9. V.A. Rokhlin, Selected topics from the metric theory of dynamical systems. Amer. Math. Soc. Transl.2 (1966) 171–240.  Zbl0185.21802
  10. P. Walters, Ergodic theory – Introductory lectures, in Lect. Notes Math.458. Springer-Verlag, Berlin (1975).  Zbl0299.28012

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