The law of series
Abstract
We prove a general ergodic-theoretic result concerning the return time statistic, which, properly understood, sheds some new light on the common sense phenomenon known as {\it the law of series}. Let \proc be an ergodic process on finitely many states, with positive entropy. We show that the distribution function of the normalized waiting time for the first visit to a small cylinder set is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function . This fact has the following interpretation: The occurrences of such a "rare event" can deviate from purely random in only one direction -- so that for any length of an "observation period" of time, the first occurrence of "attracts" its further repetitions in this period.
Keywords
Cite
@article{arxiv.math/0601166,
title = {The law of series},
author = {Tomasz Downarowicz and Yves Lacroix},
journal= {arXiv preprint arXiv:math/0601166},
year = {2007}
}