Complete convergence and records for dynamically generated stochastic processes

Freitas, Ana Cristina; Freitas, Jorge; Magalhães, Mário

doi:10.1090/tran/7922

Complete convergence and records for dynamically generated stochastic processes
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by Ana Cristina Moreira Freitas, Jorge Milhazes Freitas and Mário Magalhães PDF

Trans. Amer. Math. Soc. 373 (2020), 435-478 Request permission

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.

References

Robert J. Adler, Weak convergence results for extremal processes generated by dependent random variables, Ann. Probab. 6 (1978), no. 4, 660–667. MR 0494408
Hale Aytaç, Jorge Milhazes Freitas, and Sandro Vaienti, Laws of rare events for deterministic and random dynamical systems, Trans. Amer. Math. Soc. 367 (2015), no. 11, 8229–8278. MR 3391915, DOI 10.1090/S0002-9947-2014-06300-9
Davide Azevedo, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Fagner B. Rodrigues, Clustering of extreme events created by multiple correlated maxima, Phys. D 315 (2016), 33–48. MR 3426918, DOI 10.1016/j.physd.2015.10.002
Davide Azevedo, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Fagner B. Rodrigues, Extreme value laws for dynamical systems with countable extremal sets, J. Stat. Phys. 167 (2017), no. 5, 1244–1261. MR 3647060, DOI 10.1007/s10955-017-1767-1
Abraham Boyarsky and PawełGóra, Laws of chaos, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. Invariant measures and dynamical systems in one dimension. MR 1461536, DOI 10.1007/978-1-4612-2024-4
H. Bruin, B. Saussol, S. Troubetzkoy, and S. Vaienti, Return time statistics via inducing, Ergodic Theory Dynam. Systems 23 (2003), no. 4, 991–1013. MR 1997964, DOI 10.1017/S0143385703000026
Maria Carvalho, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, Mark Holland, and Matthew Nicol, Extremal dichotomy for uniformly hyperbolic systems, Dyn. Syst. 30 (2015), no. 4, 383–403. MR 3430306, DOI 10.1080/14689367.2015.1056722
J.-R. Chazottes and P. Collet, Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 33 (2013), no. 1, 49–80. MR 3009103, DOI 10.1017/S0143385711000897
P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 401–420. MR 1827111, DOI 10.1017/S0143385701001201
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
Dmitry Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1637–1689. MR 2034323, DOI 10.1090/S0002-9947-03-03335-X
Meyer Dwass, Extremal processes, Ann. Math. Statist. 35 (1964), 1718–1725. MR 177440, DOI 10.1214/aoms/1177700394
Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mário Magalhães, Convergence of marked point processes of excesses for dynamical systems, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 9, 2131–2179. MR 3836843, DOI 10.4171/JEMS/808
Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, Hitting time statistics and extreme value theory, Probab. Theory Related Fields 147 (2010), no. 3-4, 675–710. MR 2639719, DOI 10.1007/s00440-009-0221-y
Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, Extreme value laws in dynamical systems for non-smooth observations, J. Stat. Phys. 142 (2011), no. 1, 108–126. MR 2749711, DOI 10.1007/s10955-010-0096-4
Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, The extremal index, hitting time statistics and periodicity, Adv. Math. 231 (2012), no. 5, 2626–2665. MR 2970462, DOI 10.1016/j.aim.2012.07.029
Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics, Comm. Math. Phys. 321 (2013), no. 2, 483–527. MR 3063917, DOI 10.1007/s00220-013-1695-0
Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, and Mike Todd, Speed of convergence for laws of rare events and escape rates, Stochastic Process. Appl. 125 (2015), no. 4, 1653–1687. MR 3310360, DOI 10.1016/j.spa.2014.11.011
Jorge Milhazes Freitas, Extremal behaviour of chaotic dynamics, Dyn. Syst. 28 (2013), no. 3, 302–332. MR 3170619, DOI 10.1080/14689367.2013.806731
Jorge Milhazes Freitas, Nicolai Haydn, and Matthew Nicol, Convergence of rare event point processes to the Poisson process for planar billiards, Nonlinearity 27 (2014), no. 7, 1669–1687. MR 3232197, DOI 10.1088/0951-7715/27/7/1669
Chinmaya Gupta, Mark Holland, and Matthew Nicol, Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps, Ergodic Theory Dynam. Systems 31 (2011), no. 5, 1363–1390. MR 2832250, DOI 10.1017/S014338571000057X
Nicolai Haydn and Sandro Vaienti, The compound Poisson distribution and return times in dynamical systems, Probab. Theory Related Fields 144 (2009), no. 3-4, 517–542. MR 2496441, DOI 10.1007/s00440-008-0153-y
Nicolai T. A. Haydn and Kasia Wasilewska, Limiting distribution and error terms for the number of visits to balls in nonuniformly hyperbolic dynamical systems, Discrete Contin. Dyn. Syst. 36 (2016), no. 5, 2585–2611. MR 3485409, DOI 10.3934/dcds.2016.36.2585
Nicolai T. A. Haydn, Nicole Winterberg, and Roland Zweimüller, Return-time statistics, hitting-time statistics and inducing, Ergodic theory, open dynamics, and coherent structures, Springer Proc. Math. Stat., vol. 70, Springer, New York, 2014, pp. 217–227. MR 3213501, DOI 10.1007/978-1-4939-0419-8_{1}0
Mark Holland, Matthew Nicol, and Andrei Török, Extreme value theory for non-uniformly expanding dynamical systems, Trans. Amer. Math. Soc. 364 (2012), no. 2, 661–688. MR 2846347, DOI 10.1090/S0002-9947-2011-05271-2
Mark Holland and Mike Todd, Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 39 (2019), no. 4, 980–1001. MR 3916843, DOI 10.1017/etds.2017.56
Tailen Hsing, On the characterization of certain point processes, Stochastic Process. Appl. 26 (1987), no. 2, 297–316. MR 923111, DOI 10.1016/0304-4149(87)90183-9
T. Hsing, J. Hüsler, and M. R. Leadbetter, On the exceedance point process for a stationary sequence, Probab. Theory Related Fields 78 (1988), no. 1, 97–112. MR 940870, DOI 10.1007/BF00718038
Olav Kallenberg, Random measures, 4th ed., Akademie-Verlag, Berlin; Academic Press, Inc., London, 1986. MR 854102
John Lamperti, On extreme order statistics, Ann. Math. Statist. 35 (1964), 1726–1737. MR 170371, DOI 10.1214/aoms/1177700395
M. R. Leadbetter, Weak convergence of high level exceedances by a stationary sequence, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34 (1976), no. 1, 11–15. MR 394832, DOI 10.1007/BF00532685
M. R. Leadbetter, Georg Lindgren, and Holger Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics, Springer-Verlag, New York-Berlin, 1983. MR 691492, DOI 10.1007/978-1-4612-5449-2
Carlangelo Liverani, Benoît Saussol, and Sandro Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 671–685. MR 1695915, DOI 10.1017/S0143385799133856
Valerio Lucarini, Davide Faranda, Ana Cristina Gomes Monteiro Moreira de Freitas, Jorge Miguel Milhazes de Freitas, Mark Holland, Tobias Kuna, Matthew Nicol, Mike Todd, and Sandro Vaienti, Extremes and recurrence in dynamical systems, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2016. MR 3558780, DOI 10.1002/9781118632321
Ian Melbourne and Roland Zweimüller, Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 2, 545–556 (English, with English and French summaries). MR 3335015, DOI 10.1214/13-AIHP586
MichałMisiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 17–51. MR 623533, DOI 10.1007/BF02698686
Toshio Mori, Limit distributions of two-dimensional point processes generated by strong-mixing sequences, Yokohama Math. J. 25 (1977), no. 2, 155–168. MR 467887
S. Y. Novak, Multilevel clustering of extremes, Stochastic Process. Appl. 97 (2002), no. 1, 59–75. MR 1870720, DOI 10.1016/S0304-4149(01)00123-5
Françoise Pène and Benoît Saussol, Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing, Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2602–2626. MR 3570026, DOI 10.1017/etds.2015.28
F. Pène and B. Saussol, Spatio-temporal Poisson processes for visits to small sets, preprint, arXiv:1803.06865, 2018.
James Pickands III, The two-dimensional Poisson process and extremal processes, J. Appl. Probability 8 (1971), 745–756. MR 295453, DOI 10.1017/s0021900200114640
Yves Pomeau and Paul Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys. 74 (1980), no. 2, 189–197. MR 576270, DOI 10.1007/BF01197757
Sidney I. Resnick, Weak convergence to extremal processes, Ann. Probability 3 (1975), no. 6, 951–960. MR 428396, DOI 10.1214/aop/1176996221
Sidney I. Resnick, Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4, Springer-Verlag, New York, 1987. MR 900810, DOI 10.1007/978-0-387-75953-1
Sidney I. Resnick and David Zeber, Clustering of Markov chain exceedances, Bernoulli 19 (2013), no. 4, 1419–1448. MR 3102909, DOI 10.3150/12-BEJSP08
Marek Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), no. 1, 69–80. MR 728198, DOI 10.4064/sm-76-1-69-80
Marek Ryszard Rychlik, Another proof of Jakobson’s theorem and related results, Ergodic Theory Dynam. Systems 8 (1988), no. 1, 93–109. MR 939063, DOI 10.1017/S014338570000434X
Benoît Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math. 116 (2000), 223–248. MR 1759406, DOI 10.1007/BF02773219
Marta Tyran-Kamińska, Convergence to Lévy stable processes under some weak dependence conditions, Stochastic Process. Appl. 120 (2010), no. 9, 1629–1650. MR 2673968, DOI 10.1016/j.spa.2010.05.010
Marta Tyran-Kamińska, Weak convergence to Lévy stable processes in dynamical systems, Stoch. Dyn. 10 (2010), no. 2, 263–289. MR 2652889, DOI 10.1142/S0219493710002942