Abstract
From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For -mixing and -mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.
Highlights
In this context and for certain classes of processes, hitting and return times with respect to a given sequence of target sets converge to the exponential distribution, modelling the unpredictability of rare events
For sets A of small measure and under mixing conditions such as the ones introduced in the preceding subsection, it is expected that μ( TA > t) is approximately exponentially distributed
When we refer to return time, we mean that we study the approximation of μ A ( TA > t), that is the measure of the same event, conditioned on the points starting in A
Summary
The close relation between the Extreme Value Theory (EVT) and the statistical properties of Poincaré recurrence has been recently quite well explored. Asymptotic statistics are obtained by studying sequences of target sets An , n ≥ 1, usually of a measure shrinking to zero In this context and for certain classes of processes, hitting and return times with respect to a given sequence of target sets converge to the exponential distribution, modelling the unpredictability of rare events. To conclude about the importance of the present work as a whole, let us mention that our results are fundamental for the study of further recurrence quantities, such as the return time function [11,12] and the waiting time function [13,14], establishing a link with information theory These random variables are known to satisfy a counterpart of the famous Shannon–McMillan–Breiman theorem (asymptotic equipartition property).
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