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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Complete convergence and records for dynamically generated stochastic processes


Authors: Ana Cristina Moreira Freitas, Jorge Milhazes Freitas and Mário Magalhães
Journal: Trans. Amer. Math. Soc. 373 (2020), 435-478
MSC (2010): Primary 37A50, 60G70, 60G55, 37B20, 37A25
DOI: https://doi.org/10.1090/tran/7922
Published electronically: September 10, 2019
MathSciNet review: 4042881
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Abstract: We consider empirical multi-dimensional rare events point processes that keep track both of the time occurrence of extremal observations and of their severity, for stochastic processes arising from a dynamical system, by evaluating a given potential along its orbits. This is done both in the absence and presence of clustering. A new formula for the piling of points on the vertical direction of bi-dimensional limiting point processes, in the presence of clustering, is given, which is then generalised for higher dimensions. The limiting multi-dimensional processes are computed for systems with sufficiently fast decay of correlations. The complete convergence results are used to study the effect of clustering on the convergence of extremal processes, record time, and record values point processes. An example where the clustering prevents the convergence of the record times point process is given.


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Additional Information

Ana Cristina Moreira Freitas
Affiliation: Centro de Matemática e Faculdade de Economia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal
Email: amoreira@fep.up.pt

Jorge Milhazes Freitas
Affiliation: Centro de Matemática e Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email: jmfreita@fc.up.pt

Mário Magalhães
Affiliation: Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email: mdmagalhaes@fc.up.pt

DOI: https://doi.org/10.1090/tran/7922
Keywords: Complete convergence, point processes, extremal processes, records, clustering, hitting times
Received by editor(s): August 14, 2018
Received by editor(s) in revised form: May 30, 2019
Published electronically: September 10, 2019
Additional Notes: The first and second authors would like to thank ICTP, where the final writing of the paper took place, for the financial support and hospitality
The first and second authors were partially supported by FCT projects FAPESP/19805/2014, PTDC/MAT-CAL/3884/2014, and PTDC/MAT-PUR/28177/2017, with national funds
The third author was partially supported by FCT grant SFRH/BPD/89474/2012, which is supported by the program POPH/FSE
All authors were partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020
Article copyright: © Copyright 2019 American Mathematical Society