Mathematics – Probability
Scientific paper
2008-10-24
Mathematics
Probability
21 pages, 1 figure
Scientific paper
We consider an ergodic process on finitely many states, with positive entropy. Our first main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. That is, the occurrences of so understood "rare event" $B$ along the time axis can appear either with gap sizes of nearly exponential distribution (like in the independent Bernoulli process), or they "attract" each-other. Our second main result states that a {\it typical} ergodic process of positive entropy has the following property: the distribution functions of the normalized hitting times for the majority of cylinders $B$ of lengths $n'$ converge to zero along a \sq\ $n'$ whose upper density is 1. The occurrences of such a cylinder $B$ "strongly attract", i.e., they appear in "series" of many frequent repetitions separated by huge gaps of nearly complete absence. These results, when properly and carefully interpreted, shed some new light, in purely statistical terms, independently from physics, on a century old (and so far rather avoided by serious science) common-sense phenomenon known as {\it the law of series}, asserting that rare events in reality, once occurred, have a mysterious tendency for untimely repetitions.
Downarowicz Tomasz
Lacroix Yves
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