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

Downarowicz, Tomasz

Lacroix, Yves

English

arXiv

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 (PZ, µ, σ) 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 ma-jority 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. Note This paper resulted from studying asymptotic laws for return/hitting time statis-tics in stationary processes, a field in ergodic theory rapidly developing in the recent years (see e.g. [A-G], [C], [C-K], [D-M], [H-L-V], [L] and the reference therein). Our result significantly contributes to this area due to both its generality and strength of the assertion. After having completely written the proof, during a free-minded discussion, the authors have discovered an astonishing interpretation of the result, clear even in terms of the common sense understanding of random processes. The organization of the paper is aimed to emphasise this discovery. The consequences for the field of asymptotic laws are moved toward the end of the paper

T. Downarowicz

Y. Lacroix

CiteSeerX

THE LAW OF SERIES

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

HAL AMU

The law of series

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.236.6425

The law of series

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