# The law of series

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1 CPT - Centre de Physique Théorique - UMR 6207 2 CPT - Centre de Physique Théorique - UMR 7332

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.

Keywords : stationary random process positive entropy return time statistic hitting time statistic repelling attracting limit law the law of series

Autor: Tomasz Downarowicz - Yves Lacroix -

Fuente: https://hal.archives-ouvertes.fr/

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