Spontaneous clustering in theoretical and some empirical stationary processes

TL;DR

This paper investigates the phenomenon of clustering in stationary ergodic processes, demonstrating that strong clustering of rare events is typical in such systems, supported by theoretical proofs and an empirical marine technology example.

Contribution

It provides a theoretical proof that strong clustering of rare events is typical in ergodic processes and relates this to empirical observations in marine technology.

Findings

Clustering contradicts the expected exponential gap distribution.

Strong clustering is typical for rare events in ergodic processes.

Empirical example from marine technology illustrates clustering phenomena.

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.