The law of series

Tomasz Downarowicz; Yves Lacroix

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The law of series

Tomasz Downarowicz·Yves Lacroix

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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 $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. This fact has the following interpretation: The occurrences of such a "rare event" $B$ can deviate from purely random in only one direction -- so that for any length of an "observation period" of time, the first occurrence of $B$ "attracts" its further repetitions in this period.

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Tomasz Downarowicz, Yves Lacroix. 2006-01-09. The law of series. https://arxiv.org/abs/math/0601166

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