Level Hitting Probabilities and Extremal Indexes for Some Particular Dynamical Systems

Article
  • 18 Downloads

Abstract

We establish an exact formula for the distribution of the partial maximum sequence generated by the stationary process obtained by iterations of the Rényi map xβx mod 1, β = 2, 3, .... We thus obtain a simple proof of some asymptotic behaviour of the extremes and the values of the extremal index. A numerical application is presented.

Keywords

Rényi map Extremes Extremal index Dynamical system Scan statistics 

Mathematics Subject Classification (2010)

60K99 60G10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Addabbo T, Alioto M, Fort A, Pasini A, Rocchi S, Vignoli V (2007) A class of maximum-period non linear congruential generators derived from the Rnyi chaotic map. IEEE Trans Circuits Syst Regul Pap 54(4):816–828MathSciNetCrossRefMATHGoogle Scholar
  2. Antoniou I, Basios V, Bosco F (1996) Probabilistic control of Chaos: the −adic rényi Map under control. Int J Bifurcation Chaos 6(8):1563–1573CrossRefMATHGoogle Scholar
  3. Antoniou I, Sadovnichii VA, Shkarin SA (1999) New extended spectral decomposition of the rényi map. Phys Lett A 258:237–243MathSciNetCrossRefMATHGoogle Scholar
  4. Bateman G (1948) On the power function of the longest run as a test for randomness in a sequence of alternatives. Biometrica 35:97–112MathSciNetCrossRefMATHGoogle Scholar
  5. Ding Y, Cheng B, Jiang Z (2008) A newly-discovered GPD-GEV relationship together with comparing their models of extreme precipitations in summer. Adv Atmospheric Sci 25:507–516CrossRefGoogle Scholar
  6. Felici M, Lucarini V, Speranza A, Vitolo R (2007) Extreme value statistics of the total energy in a intermediate complexity model of the mid-latitude atmospheric jet. Part I : stationary case. J Atmos Sci 64:2159–2175CrossRefGoogle Scholar
  7. Feranda D (2016) Application of extreme value theory in dynamical systems for the analysis of blood pressure. Handbook of applications of chaos theory, Christos H. Skiadas and Charilaos Skiadas. Chapman and Hall/CRCGoogle Scholar
  8. Freitas ACM, Freitas J (2008) On the link between dependence and independence in extreme value theory for dynamical systems. Statistics & Probability Letters 8(9):1088–109MathSciNetCrossRefMATHGoogle Scholar
  9. Freitas ACM (2009) Statistics of the maximum for the tent map. Chaos, Solitons Fractals 42(1):604–608MathSciNetCrossRefMATHGoogle Scholar
  10. Freitas ACM, Freitas JM, Todd M (2012) Extremal Index, Hitting Time Statistics and periodicity. Adv Math 231:2626–2665MathSciNetCrossRefMATHGoogle Scholar
  11. Fu J (2001) Distribution of the scan statistics for a sequence of bistate trials. J Appl Probab 38:908–916MathSciNetCrossRefMATHGoogle Scholar
  12. Gaspard P (1992) r-adic one-dimensional maps and the Euler Summation Formula. J Phys A 25:L 483MathSciNetCrossRefMATHGoogle Scholar
  13. Glaz J, Naus J, Wallenstein S (2001) Scan Statistics. Springer-VerlagGoogle Scholar
  14. Gnedenko B (1943) Sur la distribution limite du terme maximum d’une suite aléatoire. Ann Math 44:423–453MathSciNetCrossRefMATHGoogle Scholar
  15. Haiman G (2003) Extreme values of the tent map process. Statistics & Probability Letters 65:451–456MathSciNetCrossRefMATHGoogle Scholar
  16. Kharin V, Zwiers F, Zhang X (2005) Intercomparison of near surface temperature and precipitation extremes in AMIP-2 simulations, reanalyses and observations. J Clim 18:5201–5233CrossRefGoogle Scholar
  17. Lasotta A, McKay M (1994) Chaos, fractals and noise, Stochastic Aspects of Dynamics. Springer- VerlagGoogle Scholar
  18. L’Ecuyer P, Simard R (2007) Testu 01 : AC library for empirical testing of random number generators. ACM Trans Math Softw 33(4):1–22MathSciNetGoogle Scholar
  19. Lucarini V, Sarno S (2011) A statistical mechanical approach for the computation of the climatic response to general forcings. Nonlinear Process Geophys 18:7–28CrossRefGoogle Scholar
  20. Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-VerlagGoogle Scholar
  21. Lucarini V, Feranda D, Wouters J, Kuna T (2014) Towards General Theory of Extreme for Observables of Chaotic Dynamical Systems. J Stat Phys 154(3):723–750MathSciNetCrossRefMATHGoogle Scholar
  22. Malvergne Y, Pisarenko V, Sornette D (2006) On the power of the generalised extreme value (GEV) and generalised Pareto distribution (GPD) estimators for empirical distributions of stock returns. Appl Financ Econ 16:271–289CrossRefGoogle Scholar
  23. de Moivre A (1967) The doctrine of chance (1738), 3rd edn. Chelsea Publishing Co., New YorkMATHGoogle Scholar
  24. Rényi A (1957) Representation for real numbers and their ergodic properties. Acta Mathematica Academiae Scientiarum Hungaricae 8:477–493MathSciNetCrossRefMATHGoogle Scholar
  25. Rokhlin V (1964) Exact endomorphisms of a Lebesgue space. Ann Math Soc Trans 39(2):1–36MATHGoogle Scholar
  26. Vannitsem S (2007) Statistical properties of the temperature maxima in an intermediate order Quasi-Geostrophic model. Tellus A 59:80–95CrossRefGoogle Scholar
  27. Vitolo R, Ruti P, Dell’Aquila A, Felici M, Lucarini V, Speranza A (2009) Accessing extremes of mid-latitudinal wave activity : methodology and application. Tellus A 61:35–49CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.MontrougeFrance

Personalised recommendations