Law of series

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 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 sequence n whose upper density

• Type: Journal
• Year: 2008
• Where: SCHOLARPEDIA
• Authors: Tomasz Downarowicz
• Keywords: Distribution Functions, Ergodic Process, Positive Entropy, SCHOLARPEDIA 2008