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Point Processes of Non stationary Sequences Generated by Sequential and Random Dynamical Systems

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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.

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Notes

  1. We remark that the sets \(U^{(\kappa )}_{j,n,i}\) depend on j (which is the maximum run of consecutive non exceedances within the same cluster) only for \(\kappa >0\). Since these \(U^{(\kappa )}_{j,n,i}\) are defined recursively, we decided to keep j in the index of the first step of the construction, \(U^{(0)}_{j,n,i}\), although strictly speaking there is no j dependence for \(\kappa =0\).

  2. \(T_\beta ^p(\zeta )=\zeta \) and p is the minimum integer with such property

  3. The result in [7], Lemma 4, is stated in a different manner. It requires \(\psi \) in \(L^{\infty }(m)\). Since the density \(h_{\omega }\) is in \(L^{\infty }(m)\) too as an element of BV(X, m),  and moreover is essentially bounded uniformly in \(\omega \) by (4.4), we get the \(\Vert \cdot \Vert _{1}\) norm on the right hand side of (4.6), as it is shown by the proof of Lemma 4 in [7].

References

  1. Aimino, R., Huyi, H., Nicol, M., Török, A., Vaienti, S.: Polynomial loss of memory for maps of the interval with a neutral fixed point. Discret. Contin. Dyn. Syst. 35(3), 793–806 (2015)

    Article  MathSciNet  Google Scholar 

  2. Aytaç, H., Freitas, J.M., Vaienti, S.: Laws of rare events for deterministic and random dynamical systems. Trans. Am. Math. Soc. 367(11), 8229–8278 (2015)

    Article  MathSciNet  Google Scholar 

  3. Berend, D., Bergelson, V.: Ergodic and mixing sequences of transformations. Ergod. Theory Dyn. Syst. 4(3), 353–366 (1984)

    Article  MathSciNet  Google Scholar 

  4. Chernick, M.R., Hsing, T., McCormick, W.P.: Calculating the extremal index for a class of stationary sequences. Adv. Appl. Probab. 23(4), 835–850 (1991)

    Article  MathSciNet  Google Scholar 

  5. Collet, P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. 21(2), 401–420 (2001)

    Article  MathSciNet  Google Scholar 

  6. Conze, J.-P., Raugi, A.: Limit Theorems for Sequential Expanding Dynamical Systems on \([0,1]\), Ergodic Theory and Related Fields, Contemporary Mathematics, vol. 430. American Mathematical Society, Providence, RI, pp. 89–121 (2007)

  7. Dragičević, D., Froyland, G., González-Tokman, C., Vaienti, S.: Almost sure invariance principle for random piecewise expanding maps. Nonlinearity 31(5), 2252–2280 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  8. Faranda, D., Alvarez-Castro, M.C., Messori, G., Rodrigues, D., Yiou, P.: The hammam effect or how a warm ocean enhances large scale atmospheric predictability. Nat. Commun. 10(1), 1316 (2019)

    Article  ADS  Google Scholar 

  9. Freitas, A.C.M., Freitas, J.M., Vaienti, S.: Extreme value laws for sequences of intermittent maps. Proc. Am. Math. Soc. 146(5), 2103–2116 (2018)

    Article  MathSciNet  Google Scholar 

  10. Freitas, A.C.M., Freitas, J.M., Todd, M.: Hitting time statistics and extreme value theory. Probab. Theory Relat. Fields 147(3–4), 675–710 (2010)

    Article  MathSciNet  Google Scholar 

  11. Freitas, A.C.M., Freitas, J.M., Todd, M.: Extreme value laws in dynamical systems for non-smooth observations. J. Stat. Phys. 142(1), 108–126 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  12. Freitas, A.C.M., Freitas, J.M., Todd, M.: The extremal index, hitting time statistics and periodicity. Adv. Math. 231(5), 2626–2665 (2012)

    Article  MathSciNet  Google Scholar 

  13. Freitas, A.C.M., Freitas, J.M., Todd, M.: The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics. Commun. Math. Phys. 321(2), 483–527 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  14. Freitas, A.C.M., Freitas, J.M., Todd, M.: Speed of convergence for laws of rare events and escape rates. Stoch. Process. Appl. 125(4), 1653–1687 (2015)

    Article  MathSciNet  Google Scholar 

  15. Freitas, A.C.M., Freitas, J.M., Vaienti, S.: Extreme value laws for non stationary processes generated by sequential and random dynamical systems. Ann. Inst. Henri Poincaré Probab. Stat. 53(3), 1341–1370 (2017)

    Article  MathSciNet  Google Scholar 

  16. Freitas, A.C.M., Freitas, J.M., Magalhães, M.: Convergence of marked point processes of excesses for dynamical systems. J. Eur. Math. Soc. (JEMS) 20(9), 2131–2179 (2018)

    Article  MathSciNet  Google Scholar 

  17. Haydn, N., Nicol, M., Török, A., Vaienti, S.: Almost sure invariance principle for sequential and non-stationary dynamical systems. Trans. Am. Math. Soc. 369(8), 5293–5316 (2017)

    Article  MathSciNet  Google Scholar 

  18. Hüsler, J.: Asymptotic approximation of crossing probabilities of random sequences. Z. Wahrsch. Verw. Gebiete 63(2), 257–270 (1983)

    Article  MathSciNet  Google Scholar 

  19. Hüsler, J.: Extreme values of nonstationary random sequences. J. Appl. Probab. 23(4), 937–950 (1986)

    Article  MathSciNet  Google Scholar 

  20. Kallenberg, O.: Random Measures, 4th edn. Akademie-Verlag, Berlin (1986)

    MATH  Google Scholar 

  21. Karr, A.F.: Point Processes and Their Statistical Inference, 2nd ed., Probability: Pure and Applied, vol. 7. Marcel Dekker, Inc., New York (1991)

  22. Kifer, Y.: Random Perturbations of Dynamical Systems. Progress in Probability and Statistics, vol. 16. Birkhäuser Boston Inc, Boston, MA (1988)

    Book  Google Scholar 

  23. Leadbetter, M.R.: On a basis for “peaks over threshold” modeling. Stat. Probab. Lett. 12(4), 357–362 (1991)

    Article  MathSciNet  Google Scholar 

  24. Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergod. Theory Dyn. Syst. 19(3), 671–685 (1999)

    Article  MathSciNet  Google Scholar 

  25. Lucarini, V., Faranda, D., Freitas, A.C.M., Freitas, J.M., Holland, M., Kuna, T., Nicol, M., Vaienti, S.: Extremes and Recurrence in Dynamical Systems. Pure and Applied Mathematics: A Wiley Series of Texts. Monographs and Tracts. Wiley, Hoboken, NJ (2016)

    Book  Google Scholar 

  26. Messori, G., Caballero, R., Faranda, D.: A dynamical systems approach to studying midlatitude weather extremes. Geophys. Res. Lett. 44(7), 3346–3354 (2017)

    Article  ADS  Google Scholar 

  27. Messori, G., Caballero, R., Bouchet, F., Faranda, D., Grotjahn, R., Harnik, N., Jewson, S., Pinto, J.G., Rivière, G., Woollings, T., Yiou, P.: An interdisciplinary approach to the study of extreme weather events: large-scale atmospheric controls and insights from dynamical systems theory and statistical mechanics. Bull. Am. Meteorol. Soc. 99(5), ES81–ES85 (2018)

    Article  ADS  Google Scholar 

  28. Nicol, M., Török, A., Vaienti, S.: Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theory Dyn. Syst. 38(3), 1127–1153 (2018)

    Article  MathSciNet  Google Scholar 

  29. Parry, W.: On the \(\beta \)-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11, 401–416 (1960)

    Article  MathSciNet  Google Scholar 

  30. Rousseau, J.: Hitting time statistics for observations of dynamical systems. Nonlinearity 27(9), 2377–2392 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  31. Rousseau, J., Todd, M.: Hitting times and periodicity in random dynamics. J. Stat. Phys. 161(1), 131–150 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  32. Rousseau, J., Saussol, B., Varandas, P.: Exponential law for random subshifts of finite type. Stoch. Process. Appl. 124(10), 3260–3276 (2014)

    Article  MathSciNet  Google Scholar 

  33. Saw, E.W., Kuzzay, D., Faranda, D., Guittonneau, A., Daviaud, F., Wiertel-Gasquet, C., Padilla, V., Dubrulle, B.: Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow. Nat. Commun. 7, 12466 EP (2016)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Centro de Matemática & Faculdade de Economia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-464, Porto, Portugal

    Ana Cristina Moreira Freitas

  2. Centro de Matemática & Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Jorge Milhazes Freitas

  3. Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Mário Magalhães

  4. Aix Marseille Université, Université de Toulon, CNRS, CPT, 13009, Marseille, France

    Sandro Vaienti

Authors
  1. Ana Cristina Moreira Freitas
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  2. Jorge Milhazes Freitas
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  3. Mário Magalhães
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  4. Sandro Vaienti
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Correspondence to Jorge Milhazes Freitas.

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Communicated by Alessandro Giuliani.

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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.

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Freitas, A.C.M., Freitas, J.M., Magalhães, M. et al. Point Processes of Non stationary Sequences Generated by Sequential and Random Dynamical Systems. J Stat Phys 181, 1365–1409 (2020). https://doi.org/10.1007/s10955-020-02630-z

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