The law of series

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 $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. That is, the occurrences of so understood "rare event" $B$ 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 $B$ of lengths $n'$ converge to zero along a \sq\ $n'$ whose upper density is 1. The occurrences of such a cylinder $B$ "strongly attract", i.e., they appear in "series" of many frequent repetitions separated by huge gaps of nearly complete absence. These results, when properly and carefully interpreted, shed some new light, in purely statistical terms, independently from physics, on a century old (and so far rather avoided by serious science) common-sense phenomenon known as {\it the law of series}, asserting that rare events in reality, once occurred, have a mysterious tendency for untimely repetitions.