Spontaneous clustering in theoretical and some empirical stationary processes

2008/10/24 by Tomasz Downarowicz, Downarowicz, Tomasz, Yves Lacroix +3
Mathematics · #37A05 #60J99 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR) #math.DS #math.PR #msc:37A05 #msc:60J99

paper · pdf · doi:10.48550/arxiv.0810.4509

9 pages, 2 figures

arxiv created 2008/10/24 · arxiv updated 2009/12/01

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.

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