Spontaneous clustering in theoretical and some empirical stationary processes*
Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzee Wyspiańskiego 27, 50-370 Wrocław, Poland
2 Institut des Sciences de l'Ingénieur de Toulon et du Var, Laboratoire Sytèmes Navals Complexes, Avenue G. Pompidou, BP 56, 83162 La Valette du Var Cedex, France
Revised: 26 August 2008
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.
Mathematics Subject Classification: 28D05 / 28D20 / 60F99 / 94A17
Key words: Stationary random process / return time / hitting time / attracting / limit law / cluster / the law of series
© EDP Sciences, SMAI, 2010