Point Processes of Non stationary Sequences Generated by Sequential and Random Dynamical Systems
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Point Processes of Non stationary Sequences Generated by Sequential and Random Dynamical Systems Ana Cristina Moreira Freitas1 · Jorge Milhazes Freitas2 Sandro Vaienti4
· Mário Magalhães3 ·
Received: 3 December 2019 / Accepted: 20 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We give general sufficient conditions to prove the convergence of marked point processes that keep record of the occurrence of rare events and of their impact for non-autonomous dynamical systems. We apply the results to sequential dynamical systems associated to both uniformly and non-uniformly expanding maps and to random dynamical systems given by fibred Lasota Yorke maps. Keywords Recurrence for non-autonomous systems · Sequential dynamical systems · Random dynamical systems · Rare events point processes Mathematics Subject Classification 37A50 · 60G70 · 60G57 · 37B20
1 Introduction The complexity of the orbital structure of chaotic systems brought special attention to the study of limiting laws of stochastic processes arising from such systems, since they borrow at least some probabilistic predictability to their erratic behaviour. The first step in this research direction is usually the construction of invariant physical measures, which provide an asymptotic spatial distribution of the orbits in the phase space and endow the stochastic processes dynamically generated with stationarity. Ergodicity then gives strong laws of large numbers. The mixing properties of the system restore asymptotic independence and, in this way, allow to MIMIC IID processes and prove limiting laws for
Communicated by Alessandro Giuliani. MM was partially supported by FCT Grant SFRH/BPD/89474/2012, which is supported by the program POPH/FSE. ACMF and JMF were partially supported by FCT Projects FAPESP/19805/2014 and PTDC/MAT-PUR/28177/2017, with national funds. All authors would like to thank the support of CMUP, which is financed by national funds through FCT, under the project with reference UIDB/00144/2020 and PTDC/MAT-CAL/3884/2014. JMF would like to thank the University of Toulon for the appointment as “Visiting Professor” during the year 2018. SV acknowledges the support of the Centro di Ricerca Matematica Ennio de Giorgi (Pisa) under the project of UniCredit Bank R&D group through the Dynamics and Information Theory Institute at the Scuola Normale Superiore. Extended author information available on the last page of the article
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the mean, such as: central limit theorems, large deviation principles, invariance principles, among others. However, in many occasions the exact formula for the invariant measure is not available and one has to rely on reference measures with respect to which these processes are not stationary anymore. Loosening stationarity leads to non-autonomous dynamical systems for which the study of limit theorems is just at the beginning. We mention the recent works [1,17,28] and references therein. While the limiting laws mentioned so far pertain to the mean or a
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