Hitting time statistics and extreme value theory

Probability Theory and Related Fields

Volumn 147, Issue 3, 2010, Pages 675-710

  Freitas, Ana Cristina Moreira ^a   Freitas, Jorge Milhazes ^a   Todd, Mike ^a  

^a UNIVERSITY OF PORTO   (Portugal)

Author keywords

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

Indexed keywords

EID: 77951104031     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-009-0221-y     Document Type: Article

References (37)

1. Abadi, M.: Sharp error terms and necessary conditions for exponential hitting times in mixing processes. Ann. Probab. 32, 243-264 (2004)
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)
3. Alves, J.F.: Strong statistical stability of non-uniformly expanding maps. Nonlinearity 17, 1193-1215 (2004)
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)
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)
6. 2 on (-1, 1). Ann. Math. 122, 1-25 (1985)
7. Bruin, H., Saussol, B., Troubetzkoy, S., Vaienti, S.: Return time statistics via inducing. Ergod. Theory Dyn. Syst. 23, 991-1013 (2003)
8. Bruin, H., Todd, M.: Return time statistics for invariant measures for interval maps with positive Lyapunov exponent. Stoch. Dyn. 9, 81-100 (2009)
9. Bruin, H., Vaienti, S.: Return time statistics for unimodal maps. Fund. Math. 176, 77-94 (2003)
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)
11. Coelho, Z.: The loss of tightness of time distributions for homeomorphisms of the circle. Trans. Am. Math. Soc. 356, 4427-4445 (2004)
12. Coelho, Z., de Faria, E.: Limit laws of entrance times for homeomorphisms of the circle. Israel J. Math. 93, 93-112 (1996)
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)
14. Collet, P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Theory Dyn. Syst. 21, 401-420 (2001)
15. Denker, M., Gordin, M., Sharova, A.A.: Poisson limit theorem for toral automorphisms. Ill. J. Math. 48, 1-20 (2004)
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)
17. Dolgopyat, D.: Limit theorems for partially hyperbolic systems. Trans. Am. Math. Soc. 356, 1637-1689 (2004)
18. Dudewicz, E.J., Mishra, S.N.: Modern Mathematical Statistics.Wiley Series in Probability and Mathematical Statistics. Wiley, London (1988)
19. Feller, W.: An introduction to Probability Theory and its Applications, vol. I. Wiley Publications in Statistics, New York (1952)
20. Freitas, A.C.M., Freitas, J.M.: Extreme values for Benedicks Carleson maps. Ergod. Theory Dyn. Syst. 28(4), 1117-1133 (2008)
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)
22. Gouëzel, S.: Decay of correlations for nonuniformly expanding systems. Bull. Soc. Math. France 134, 1-31 (2006)
23. Haydn, N., Lacroix,Y.,Vaienti, S.: Hitting and return times in ergodic dynamical systems. Ann. Probab. 33, 2043-2050 (2005)
24. Haydn, N., Vaienti, S.: The compound Poisson distribution and return times in dynamical systems. Probab. Theory Relat. Fields 144, 517-542 (2009)
25. Hirata, M.: Poisson law for axiom a diffeomorphisms. Ergod.Theory Dyn. Syst. 13, 533-556 (1993)
26. Hirata, M., Saussol, B., Vaienti, S.: Statistics of return times: a general framework and new applications. Comm. Math. Phys. 206, 33-55 (1999)
27. Holland, M., Nicol, M., Torok, A.: Extreme value distributions for non-uniformly hyperbolic dynamical systems (preprint) http://www.math.uh.edu/~nicol/papers.html
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)
29. Kallenberg, O.: Random Measures. Academic Press Inc., New York (1986)
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)
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)
32. Pesin, Y.: Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago (1997)
33. Pitskel, B.: Poisson limit law for Markov chains. Ergod. Theory Dynam. Syst. 11, 501-513 (1991)
34. Rychlik, M.: Bounded variation and invariant measures. Stud. Math. 76, 69-80 (1983)
35. Young, L.S.: Decay of correlations for certain quadratic maps.Comm. Math. Phys. 146, 123-138 (1992)
36. Young, L.S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(2), 585-650 (1998)
37. Young, L.S.: Recurrence times and rates of mixing. Isr. J. Math. 110, 153-188 (1999)