Extreme Value Theory for Long-Range-Dependent Stable Random Fields

  • Zaoli Chen
  • Gennady SamorodnitskyEmail author


We study the extremes for a class of a symmetric stable random fields with long-range dependence. We prove functional extremal theorems both in the space of sup measures and in the space of càdlàg functions of several variables. The limits in both types of theorems are of a new kind, and only in a certain range of parameters, these limits have the Fréchet distribution.


Random field Extremal limit theorem Random sup measure Random closed set Long-range dependence Stable law Heavy tails 

Mathematics Subject Classification (2010)

Primary 60G60 60G70 60G52 



  1. 1.
    Aaronson, J.: An Introduction to Infinite Ergodic Theory, volume 50 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)Google Scholar
  2. 2.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)CrossRefGoogle Scholar
  3. 3.
    Chakrabarty, A., Roy, P.: Group theoretic dimension of stationary symmetric \(\alpha \)-stable random fields. J. Theor. Probab. 26, 240–258 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer, New York (2001)CrossRefGoogle Scholar
  5. 5.
    de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer, New York (2006)CrossRefGoogle Scholar
  6. 6.
    Fisher, R.A., Tippett, L.: Limiting forms of the frequency distributions of the largest or smallest member of a sample. Proc. Camb. Philis. Soc. 24, 180–190 (1928)CrossRefGoogle Scholar
  7. 7.
    Fitzsimmons, P., Taksar, M.: Stationary regenerative sets and subordinators. Ann. Probab. 16, 1299–1305 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gnedenko, B.: Sur la distribution limite du terme maximum d’une serie aleatoire. Ann. Math. 44, 423–453 (1943)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lacaux, C., Samorodnitsky, G.: Time-changed extremal process as a random sup measure. Bernoulli 22, 1979–2000 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Leadbetter, M., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)CrossRefGoogle Scholar
  11. 11.
    Leadbetter, M., Rootzén, H.: On Extremes Values in Stationary Random Fields. In Stochastic Processes and related Topics, pp. 275–285. Birkhäuser, Boston (1998)CrossRefGoogle Scholar
  12. 12.
    Molchanov, I.: Theory of Random Sets, 2nd edn. Springer, London (2017)CrossRefGoogle Scholar
  13. 13.
    Norberg, T.: On Vervaat’s sup vague topology. Arkiv för Matematik 28, 139–144 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nualart, D.: The Malliavin Calculus and Related Topics. Springer, New York (1995)CrossRefGoogle Scholar
  15. 15.
    O’Brien, G., Torfs, P., Vervaat, W.: Stationary self-similar extremal processes. Probab. Theory Relat. Fields 87, 97–119 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Owada, T.: Limit theory for the sample autocovariance for heavy tailed stationary infinitely divisible processes generated by conservative flows. J. Theor. Probab. 29, 63–95 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Owada, T., Samorodnitsky, G.: Functional Central Limit Theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43, 240–285 (2015a)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Owada, T., Samorodnitsky, G.: Maxima of long memory stationary symmetric \(\alpha \)-stable processes, and self-similar processes with stationary max-increments. Bernoulli 21, 1575–1599 (2015b)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Resnick, S.: Extreme Values, Regular Variation and Point Processes. Springer, New York (1987)CrossRefGoogle Scholar
  20. 20.
    Resnick, S., Samorodnitsky, G., Xue, F.: Growth rates of sample covariances of stationary symmetric \(\alpha \)-stable processes associated with null recurrent Markov chains. Stoch. Process. Appl. 85, 321–339 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rosiński, J.: Decomposition of stationary \(\alpha \)-stable random fields. Ann. Probab. 28, 1797–1813 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Roy, P., Samorodnitsky, G.: Stationary symmetric \(\alpha \)-stable discrete parameter random fields. J. Theor. Probab. 21, 212–233 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Salinetti, G., Wets, R.: On the convergence of closed-valued measurable multifunctions. Trans. Am. Math. Soc. 266, 275–289 (1981)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Samorodnitsky, G.: Extreme value theory, ergodic theory, and the boundary between short memory and long memory for stationary stable processes. Ann. Probab. 32, 1438–1468 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Samorodnitsky, G.: Stochastic Processes and Long Range Dependence. Springer, Cham (2016)CrossRefGoogle Scholar
  26. 26.
    Samorodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman and Hall, New York (1994)zbMATHGoogle Scholar
  27. 27.
    Samorodnitsky, G., Wang, Y.: Extremal theory for long range dependent infinitely divisible processes. Technical report (2017)Google Scholar
  28. 28.
    Sarkar, S., Roy, P.: Stable random fields indexed by finitely generated free groups. Ann. Probab. 46, 2680–2714 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  30. 30.
    Straf, M.: Weak convergence of stochastic processes with several parameters. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, volume 2. University of California Press, Berkeley, CA, pp. 187–221 (1972)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

Personalised recommendations