The law of series
Tomasz Downarowicz, Yves Lacroix
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 the law of series. Let 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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