Hitting Times and Positions in Rare Events
Annales Henri Lebesgue, Volume 5 (2022), pp. 1361-1415.

We establish abstract limit theorems which provide sufficient conditions for a sequence (A l ) of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the A l to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to A l before some time τ l which is of order o(1/μ(A l )). In particular, no assumptions on the asymptotic behavior of the system akin to classical mixing conditions are used. We also discuss some general questions about the asymptotic behaviour of spatial and spatiotemporal processes, and illustrate our results in a setup of simple prototypical systems.

Nous établissons des théorèmes limites abstraits qui fournissent des conditions suffisantes pour qu’une suite (A l ) d’événements rares dans un système préservant une mesure de probabilité ergodique satisfasse des asymptotiques de Poisson, et pour que les positions dans A l soient asymptotiquement iid (limites de Poisson spatio-temporelles). Les théorèmes limites n’utilisent que des informations sur ce qui arrive à A l avant un certain temps τ l d’ordre o(1/μ(A l )). En particulier, nous n’utilisons aucune hypothèse sur le comportement asymptotique du système du type conditions de mélange classiques. Nous discutons également quelques questions générales sur le comportement asymptotique des processus spatiaux et spatio-temporels, et illustrons nos résultats avec des systèmes prototypiques simples.

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DOI: 10.5802/ahl.154
Classification: 28D05, 37A25, 37A50, 37C30, 60F05, 11K50
Keywords: invariant measure, limit distribution, rare events, Poisson process
Zweimüller, Roland 1

1 Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna (Austria)
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Zweimüller, Roland. Hitting Times and Positions in Rare Events. Annales Henri Lebesgue, Volume 5 (2022), pp. 1361-1415. doi : 10.5802/ahl.154. http://www.numdam.org/articles/10.5802/ahl.154/

[Aar97] Aaronson, Jon An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, 1997 | DOI | Zbl

[Aba04] Abadi, Miguel Sharp error terms and necessary conditions for exponential hitting times in mixing processes, Ann. Probab., Volume 32 (2004) no. 1A, pp. 243-264 | MR | Zbl

[AS11] Abadi, Miguel; Saussol, Benoît Hitting and returning to rare events for all alpha-mixing processes, Stochastic Processes Appl., Volume 121 (2011) no. 2, pp. 314-323 | DOI | MR | Zbl

[Bil86] Billingsley, Patrick Probability and Measure, Wiley Series in Probability and Mathematical Statistics: Probability and mathematical statistics, John Wiley & Sons, 1986 | Zbl

[Bil99] Billingsley, Patrick Convergence of Probability Measures, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999 | DOI | Zbl

[BSTV03] Bruin, Henk; Saussol, Benoît; Troubetzkoy, Serge E.; Vaienti, Sandro Return time statistics via inducing, Ergodic Theory Dyn. Syst., Volume 23 (2003) no. 4, pp. 991-1013 | DOI | MR | Zbl

[CC13] Chazottes, Jean-René; Collet, Pierre Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems, Ergodic Theory Dyn. Syst., Volume 33 (2013) no. 1, pp. 49-80 | DOI | MR | Zbl

[Doe40] Doeblin, Wolfgang Remarques sur la théorie métrique des fractions continues, Compos. Math., Volume 7 (1940), pp. 353-371 | Numdam | Zbl

[Eag76] Eagleson, G. K. Some simple conditions for limit theorems to be mixing, Teor. Veroyatn. Primen., Volume 21 (1976), pp. 653-660 | MR | Zbl

[FFT12] Freitas, Ana Cristina M.; Freitas, Jorge M.; Todd, Mike The extremal index, hitting time statistics and periodicity, Adv. Math., Volume 231 (2012) no. 5, pp. 2626-2665 | DOI | MR | Zbl

[FFTV16] Freitas, Ana Cristina M.; Freitas, Jorge M.; Todd, Mike; Vaienti, Sandro Rare events for the Manneville–Pomeau map, Stochastic Processes Appl., Volume 126 (2016) no. 11, pp. 3463-3479 | DOI | MR | Zbl

[Gau12] Gauss, Carl F. Letter to Laplace (1812) (Göttingen, January 30 th )

[HLV05] Haydn, Nicolai T.; Lacroix, Yves; Vaienti, Sandro Hitting and return times in ergodic dynamical systems, Ann. Probab., Volume 33 (2005) no. 5, pp. 2043-2050 | MR | Zbl

[Hol05] Holland, Mark Slowly mixing systems and intermittency maps, Ergodic Theory Dyn. Syst., Volume 25 (2005) no. 1, pp. 133-159 | DOI | MR | Zbl

[HP14] Haydn, Nicolai T.; Psiloyenis, Y. Return times distribution for Markov towers with decay of correlations, Nonlinearity, Volume 27 (2014) no. 6, pp. 1323-1349 | DOI | MR | Zbl

[HSV99] Hirata, Masaki; Saussol, Benoît; Vaienti, Sandro Statistics of Return Times: A General Framework and New Applications, Commun. Math. Phys., Volume 206 (1999) no. 1, pp. 33-55 | DOI | MR | Zbl

[HWZ14] Haydn, Nicolai T.; Winterberg, Nicole; Zweimüller, Roland Return-time statistics, Hitting-time statistics and Inducing, Ergodic Theory, Open Dynamics, and Coherent Structures (Springer Proceedings in Mathematics & Statistics), Volume 70, Springer, 2014, pp. 217-227 | DOI | MR | Zbl

[HY16] Haydn, Nicolai T.; Yang, Fan Entry times distribution for mixing systems, J. Stat. Phys., Volume 163 (2016) no. 2, pp. 374-392 | DOI | MR | Zbl

[IK02] Iosifescu, Marius; Kraaikamp, Cor Metrical Theory of Continued Fractions, Mathematics and its Applications (Dordrecht), 547, Kluwer Academic Publishers, 2002 | DOI | Zbl

[Ios77] Iosifescu, Marius A Poisson law for ψ-mixing sequences establishing the truth of a Doeblin’s statement, Rev. Roum. Math. Pures Appl., Volume 22 (1977), pp. 1441-1447 | MR | Zbl

[Kre85] Krengel, Ulrich Ergodic Theorems, De Gruyter Studies in Mathematics, 6, Walter de Gruyter, 1985 | DOI | Zbl

[Mar17] Marklof, Jens Entry and return times for semi-flows, Nonlinearity, Volume 30 (2017) no. 2, pp. 810-824 | DOI | MR | Zbl

[PS16] Pène, Françoise; Saussol, Benoît Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing, Ergodic Theory Dyn. Syst., Volume 36 (2016) no. 8, pp. 2602-2626 | DOI | MR | Zbl

[PS20] Pène, Françoise; Saussol, Benoît Spatio-temporal Poisson processes for visits to small sets, Isr. J. Math., Volume 240 (2020) no. 2, pp. 625-665 | DOI | MR | Zbl

[PSZ17] Pène, Françoise; Saussol, Benoît; Zweimüller, Roland Return- and hitting-time limits for rare events of null-recurrent Markov maps, Ergodic Theory Dyn. Syst., Volume 37 (2017) no. 1, pp. 244-276 | DOI | MR | Zbl

[PT20] Pène, Françoise; Thomine, Damien Potential kernel, hitting probabilities and distributional asymptotics, Ergodic Theory Dyn. Syst., Volume 40 (2020) no. 7, pp. 1894-1967 | DOI | MR | Zbl

[Res08] Resnick, Sidney I. Extreme Values, Regular Variation and Point Processes, Springer Series in Operations Research and Financial Engineering, Springer, 2008 | Zbl

[Roh64] Rohlin, Vladimir A. Exact endomorphisms of a Lebesgue space, Am. Math. Soc., Transl., II. Ser., Volume 39 (1964), pp. 1-36 | DOI | Zbl

[RZ20] Rechberger, Simon; Zweimüller, Roland Return- and hitting-time distributions of small sets in infinite measure preserving systems, Ergodic Theory Dyn. Syst., Volume 40 (2020) no. 8, pp. 2239-2273 | DOI | MR | Zbl

[Rén58] Rényi, Alfréd On mixing sequences of sets, Acta Math. Acad. Sci. Hung., Volume 9 (1958), pp. 215-228 | DOI | MR | Zbl

[Tha80] Thaler, Maximilian Estimates of the invariant densities of endomorphisms with indifferent fixed points, Isr. J. Math., Volume 37 (1980), pp. 303-314 | DOI | MR | Zbl

[Tha05] Thaler, Maximilian Asymptotic distributions and large deviations for iterated maps with an indifferent fixed point, Stoch. Dyn., Volume 5 (2005) no. 3, pp. 425-440 | DOI | MR | Zbl

[TK10] Tyran-Kamińska, Marta Weak convergence to Lévy stable processes in dynamical systems, Stoch. Dyn., Volume 10 (2010) no. 2, pp. 263-289 | DOI | Zbl

[Whi02] Whitt, Ward Stochastic-Process Limits. An introduction to stochastic-process limits and their application to queue, Springer Series in Operations Research, Springer, 2002 | DOI | Zbl

[Yos38] Yosida, Kosaku Mean ergodic theorem in Banach spaces, Proc. Imp. Acad. Japan, Volume 14 (1938), pp. 292-294 | MR | Zbl

[Zwe03] Zweimüller, Roland Stable limits for probability preserving maps with indifferent fixed points, Stoch. Dyn., Volume 3 (2003) no. 1, pp. 83-99 | DOI | MR | Zbl

[Zwe07a] Zweimüller, Roland Infinite measure preserving transformations with compact first regeneration, J. Anal. Math., Volume 103 (2007), pp. 93-131 | DOI | MR | Zbl

[Zwe07b] Zweimüller, Roland Mixing limit theorems for ergodic transformations, J. Theor. Probab., Volume 20 (2007) no. 4, pp. 1059-1071 | DOI | MR | Zbl

[Zwe16] Zweimüller, Roland The general asymptotic return-time process, Isr. J. Math., Volume 212 (2016) no. 1, pp. 1-36 | DOI | MR | Zbl

[Zwe19] Zweimüller, Roland Hitting-time limits for some exceptional rare events of ergodic maps, Stochastic Processes Appl., Volume 129 (2019) no. 5, pp. 1556-1567 | DOI | MR | Zbl

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