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The law of series

Published online by Cambridge University Press:  22 April 2010

T. DOWNAROWICZ  and
Y. LACROIX
T. DOWNAROWICZ
Affiliation:
Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland (email: downar@pwr.wroc.pl, downar@im.pwr.wroc.pl)
Y. LACROIX
Affiliation:
Institut des Sciences de l’Ingénieur de Toulon et du Var, Avenue G. Pompidou, B.P. 56, 83162 La Valette du Var Cedex, France (email: yves.lacroix@univ-tln.fr)
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Abstract

We consider an ergodic process on finitely many states, with positive entropy. Our main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set B is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function 1−e−t. That is, the occurrences of B along the time axis can appear either with gap sizes of nearly the exponential distribution (as in an independent and identically distributed process), or they attract each other and create ‘series’. We recall that in [T. Downarowicz, Y. Lacroix and D. Leandri. Spontaneous clustering in theoretical and some empirical stochastic processes. ESAIM Probab. Stat. to appear] it is proved that in a typical (in the sense of category) ergodic process (of any entropy), all cylinders B of selected lengths (such lengths have upper density 1 in ℕ) reveal strong attracting. Combining this with the result of this paper, we obtain, globally in ergodic processes of positive entropy and for long cylinder sets, the prevalence of attracting and deficiency of repelling. This phenomenon resembles what in real life is known as the law of series; the common-sense observation that a rare event, having occurred, has a mysterious tendency to untimely repetitions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 351 - 367
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Abadi, M. and Galves, A.. Inequalities for the occurrence times of rare events in mixing processes. The state of the art. Inhomogeneous random systems. Markov Process. Related Fields 7 (2001), 97–112.Google Scholar
[2]Chaumoître, V. and Kupsa, M.. Asymptotics for return times of rank one systems. Stoch. Dyn. 5 (2005), 65–73.CrossRefGoogle Scholar
[3]Coelho, Z.. Asymptotic laws for symbolic dynamical systems. Topics in Symbolic Dynamics and Applications (London Mathematical Society Lecture Note Series, 279). Cambridge University Press, Cambridge, 2000, pp. 123–165.Google Scholar
[4]Durand, F. and Maass, A.. Limit laws for entrance times for low-complexity Cantor minimal systems. Nonlinearity 14 (2001), 683–700.CrossRefGoogle Scholar
[5]Downarowicz, T., Lacroix, Y. and Leandri, D.. Spontaneous clustering in theoretical and some empirical stochastic processes. ESAIM Probab. Stat. to appear.Google Scholar
[6]Haydn, N., Lacroix, Y. and Vaienti, S.. Hitting and return times in ergodic dynamical systems. Ann. Probab. 33 (2005), 2043–2050.CrossRefGoogle Scholar
[7]Hirata, N., Saussol, B. and Vaienti, S.. Statistics of return times: a general framework and new applications. Commun. Math. Phys. 206 (1999), 33–55.CrossRefGoogle Scholar
[8]Kac, M.. On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53 (1947), 1002–1010.CrossRefGoogle Scholar
[9]Lacroix, Y.. Possible limit laws for entrance times of an ergodic aperiodic dynamical system. Israel J. Math. 132 (2002), 253–263.CrossRefGoogle Scholar
[10]Ornstein, D. S. and Weiss, B.. Entropy and recurrence rates for stationary random fields. IEEE Trans. Inform. Theory 48 (2002), 1694–1697.CrossRefGoogle Scholar
[11]Petersen, K.. Ergodic Theory. Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
[12]Rudolph, D. J.. If a two-point extension of a Bernoulli shift has an ergodic square, then it is Bernoulli. Israel J. Math. 30 (1978), 159–180.CrossRefGoogle Scholar
[13]Shields, P.. The Theory of Bernoulli Shifts (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, 1973.Google Scholar
[14]Smorodinsky, M.. On Ornstein’s isomorphism theorem for Bernoulli shifts. Adv. Math. 9 (1972), 1–9.CrossRefGoogle Scholar
[15]Walters, P.. Ergodic Theory—Introductory Lectures (Lecture Notes in Mathematics, 458). Springer, Berlin, 1975.CrossRefGoogle Scholar