The law of series
Tomasz Downarowicz, Yves Lacroix

TL;DR
This paper investigates the statistical behavior of rare events in ergodic processes with positive entropy, revealing that such events tend to cluster or appear in series, providing a new statistical perspective on the law of series.
Contribution
It establishes that in ergodic processes, rare events either follow an exponential waiting time distribution or tend to cluster in series, offering new insights into the law of series.
Findings
Most rare events have waiting times dominated by exponential distribution.
Typical processes show rare events occurring in series with large gaps.
Results provide a statistical explanation for the law of series phenomenon.
Abstract
We consider an ergodic process on finitely many states, with positive entropy. Our first 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 is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function . That is, the occurrences of so understood "rare event" along the time axis can appear either with gap sizes of nearly exponential distribution (like in the independent Bernoulli process), or they "attract" each-other. Our second main result states that a {\it typical} ergodic process of positive entropy has the following property: the distribution functions of the normalized hitting times for the majority of cylinders of lengths converge to zero along a \sq\ whose upper density is 1. The occurrences of such a…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsProbability and Statistical Research
