Probability Theory and Related Fields

, Volume 147, Issue 3–4, pp 675–710 | Cite as

Hitting time statistics and extreme value theory

  • Ana Cristina Moreira Freitas
  • Jorge Milhazes FreitasEmail author
  • Mike Todd


We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We also give applications of our theory to higher dimensional examples, for which we also obtain classical extreme value laws and exponential hitting time statistics (for balls). We extend these ideas to the subsequent returns to asymptotically small sets, linking the Poisson statistics of both processes.


Return time statistics Extreme value theory Non-uniform hyperbolicity Interval maps 

Mathematics Subject Classification (2000)

37A50 37C40 60G10 60G70 37B20 37D25 37E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abadi M.: Sharp error terms and necessary conditions for exponential hitting times in mixing processes. Ann. Probab. 32, 243–264 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abadi M., Galves A.: Inequalities for the occurrence times of rare events in mixing processes. The state of the art. Markov Process. Relat. Fields 7, 97–112 (2001)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Alves J.F.: Strong statistical stability of non-uniformly expanding maps. Nonlinearity 17, 1193–1215 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alves J.F., Bonatti C., Viana M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140, 351–398 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alves J.F., Luzzatto S., Pinheiro V.: Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 817–839 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Benedicks M., Carleson L.: On iterations of 1 − ax 2 on (−1, 1). Ann. Math. 122, 1–25 (1985)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bruin H., Saussol B., Troubetzkoy S., Vaienti S.: Return time statistics via inducing. Ergod. Theory Dyn. Syst. 23, 991–1013 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bruin H., Todd M.: Return time statistics for invariant measures for interval maps with positive Lyapunov exponent. Stoch. Dyn. 9, 81–100 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bruin H., Vaienti S.: Return time statistics for unimodal maps. Fund. Math. 176, 77–94 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Coelho, Z.: Asymptotic laws for symbolic dynamical systems. Topics in Symbolic Dynamics and Applications (Temuco 1997). LMS Lecture Notes Series, vol. 279, pp. 123–165. Cambridge University Press, London (2000)Google Scholar
  11. 11.
    Coelho Z.: The loss of tightness of time distributions for homeomorphisms of the circle. Trans. Am. Math. Soc. 356, 4427–4445 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Coelho Z., de Faria E.: Limit laws of entrance times for homeomorphisms of the circle. Israel J. Math. 93, 93–112 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Collet, P.: Some ergodic properties of maps of the interval. Dynamical Systems (Temuco, 1991/1992), (Travaux en cours, 52), pp. 55–91. Herman, Paris (1996)Google Scholar
  14. 14.
    Collet P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. 21, 401–420 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Denker M., Gordin M., Sharova A.A.: Poisson limit theorem for toral automorphisms. Ill. J. Math. 48, 1–20 (2004)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Denker M., Philipp W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Theory Dyn. Syst. 4, 541–552 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Dolgopyat D.: Limit theorems for partially hyperbolic systems. Trans. Am. Math. Soc. 356, 1637–1689 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dudewicz, E.J., Mishra, S.N.: Modern Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, London (1988)Google Scholar
  19. 19.
    Feller W.: An introduction to Probability Theory and its Applications, vol. I. Wiley Publications in Statistics, New York (1952)Google Scholar
  20. 20.
    Freitas A.C.M., Freitas J.M.: Extreme values for Benedicks Carleson maps. Ergod. Theory Dyn. Syst. 28(4), 1117–1133 (2008)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Freitas A.C.M., Freitas J.M.: On the link between dependence and independence in extreme value theory for dynamical systems. Stat. Probab. Lett. 78, 1088–1093 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Gouëzel S.: Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134, 1–31 (2006)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Haydn N., Lacroix Y., Vaienti S.: Hitting and return times in ergodic dynamical systems. Ann. Probab. 33, 2043–2050 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Haydn N., Vaienti S.: The compound Poisson distribution and return times in dynamical systems. Probab. Theory Relat. Fields 144, 517–542 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hirata M.: Poisson law for axiom a diffeomorphisms. Ergod.Theory Dyn. Syst. 13, 533–556 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hirata M., Saussol B., Vaienti S.: Statistics of return times: a general framework and new applications. Comm. Math. Phys. 206, 33–55 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Holland, M., Nicol, M., Torok, A.: Extreme value distributions for non-uniformly hyperbolic dynamical systems (preprint)
  28. 28.
    Hsing T., Hüsler J., Leadbetter M.R.: On the exceedance point process for a stationary sequence. Probab. Theory Relat. Fields 78, 97–112 (1988)zbMATHCrossRefGoogle Scholar
  29. 29.
    Kallenberg O.: Random Measures. Academic Press Inc., New York (1986)Google Scholar
  30. 30.
    Keller G., Nowicki T.: Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Comm. Math. Phys. 149, 31–69 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Lindgren, G., Leadbetter, M.R., Rootzén, H.: Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer, New York (1983)Google Scholar
  32. 32.
    Pesin Y.: Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago (1997)Google Scholar
  33. 33.
    Pitskel B.: Poisson limit law for Markov chains. Ergod. Theory Dynam. Syst. 11, 501–513 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Rychlik M.: Bounded variation and invariant measures. Stud. Math. 76, 69–80 (1983)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Young L.S.: Decay of correlations for certain quadratic maps. Comm. Math. Phys. 146, 123–138 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Young L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(2), 585–650 (1998)zbMATHCrossRefGoogle Scholar
  37. 37.
    Young L.S.: Recurrence times and rates of mixing. Isr. J. Math. 110, 153–188 (1999)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Ana Cristina Moreira Freitas
    • 1
  • Jorge Milhazes Freitas
    • 2
    Email author
  • Mike Todd
    • 2
  1. 1.Faculdade de Economia, Centro de MatemáticaUniversidade do PortoPortoPortugal
  2. 2.Centro de Matemática da Universidade do PortoPortoPortugal

Personalised recommendations