, Volume 20, Issue 3, pp 493–517 | Cite as

Generalized Pickands constants and stationary max-stable processes

  • Krzysztof DȩbickiEmail author
  • Sebastian Engelke
  • Enkelejd Hashorva


Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (Bernoulli, 20(3), 1600–1619, 2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker–Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore develop a link to extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.


Brown–Resnick process Fractional Brownian motion Gaussian process Generalized Pickands constant Lévy process Max-stable process Mixed moving maxima representation 

AMS 2000 Subject Classifications

60G15 60G70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We are grateful to three anonymous referees, Thomas Mikosch and Ilya Molchanov for numerous important suggestions. In particular, the new M3 representation (??) was suggested by one of the referees. Financial support by the Swiss National Science Foundation grants 200021-166274 (EH) and 161297 (SE), and partial support by NCN Grant No 2015/17/B/ST1/01102 (2016-2019) (KD) is gratefully acknowledged.


  1. Aldous, D.: Probability Approximations via the Poisson Clumping Heuristic, Volume 77 of Applied Mathematical Sciences. Springer, New York (1989). ISBN 0-387-96899-7. doi: 10.1007/978-1-4757-6283-9
  2. Asmussen, S., Albrecher, H.: Ruin Probabilities, vol. 14. World Scientific (2010)Google Scholar
  3. Berman, S.M.: Sojourns and extremes of stochastic processes. The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove CA (1992)Google Scholar
  4. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)zbMATHGoogle Scholar
  5. Davis, R.A., Mikosch, T., Zhao, Y.: Measures of serial extremal dependence and their estimation. Stochastic Process. Appl. 123(7), 2575–2602 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dȩbicki, K., Kisowski, P.: A note on upper estimates for Pickands constants. Statistics & Probability Letters 78(14), 2046–2051 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dȩbicki, K.: Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98(1), 151–174 (2002)MathSciNetCrossRefGoogle Scholar
  8. Dȩbicki, K.: Some properties of generalized Pickands constants. Teor. Veroyatn. Primen. 50(2), 396–404 (2005)MathSciNetCrossRefGoogle Scholar
  9. Dȩbicki, K., Mandjes, M.: Queues and Lévy Fluctuation Theory. Springer (2015)Google Scholar
  10. Dȩbicki, K., Michna, Z., Rolski, T.: Simulation of the asymptotic constant in some fluid models. Stoch. Models 19(3), 407–423 (2003a). doi: 10.1081/STM-120023567. ISSN 1532-6349MathSciNetCrossRefzbMATHGoogle Scholar
  11. Dȩbicki, K., Michna, Z., Rolski, T.: Simulation of the asymptotic constant in some fluid models. Stoch. Model. 19(3), 407–423 (2003b)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Delorme, M., Rosso, A., Wiese, K.J.: Pickands’ Constant at First Order in an Expansion Around Brownian Motion. arXiv:1609.07909[cond-mat.stat-mech] (2016)
  13. Dieker, A.B., Mikosch, T.: Exact simulation of Brown-Resnick random fields at a finite number of locations. Extremes 18, 301–314 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dieker, A.B., Yakir, B.: On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20(3), 1600–1619 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Dombry, C., Kabluchko, Z.: Ergodic decompositions of stationary max-stable processes in terms of their spectral functions. arXiv:1601.00792[math.PR] (2016)
  16. Dombry, C., Engelke, S., Oesting, M.: Exact simulation of max-stable processes. Biometrika 103, 303–317 (2016)MathSciNetCrossRefGoogle Scholar
  17. Engelke, S., Ivanovs, J.: A Lévy-derived process seen from its supremum and max-stable processes. Electron. J. Probab., 21 (2016)Google Scholar
  18. Engelke, S., Kabluchko, Z.: Max-stable processes associated with stationary systems of independent lévy particles. Stochastic Process. Appl. 125(11), 4272–4299 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Engelke, S., Kabluchko, Z., Schlather, M.: An equivalent representation of the Brown-Resnick process. Statist. Probab. Lett. 81, 1150–1154 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Engelke, S., Malinowski, A., Oesting, M., Schlather, M.: Statistical inference for max-stable processes by conditioning on extreme events. Adv. Appl. Probab. 46 (2), 478–495 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Engelke, S., Kabluchko, Z., Schlather, M.: Maxima of independent, non-identically distributed Gaussian vectors. Bernoulli 21, 38–61 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Feinberg, E.A., Kasyanov, P.O., Zadoianchuk, N.V.: Fatou’s lemma for weakly converging probabilities. Theory Probab. Appl. 58(4), 683–689 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Harper, A.J.: Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab. 23(2), 584–616 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Harper, A.J.: Pickands’ constant h α does not equal 1/γ(1/α), for small α. Bernoulli accepted (2015)Google Scholar
  25. Hüsler, J., Piterbarg, V.I.: Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83(2), 257–271 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Kabluchko, Z.: Spectral representations of sum- and max-stable processes. Extremes 12, 401–424 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Kabluchko, Z.: Extremes of independent Gaussian processes. Extremes 14, 285–310 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kabluchko, Z., Wang, Y.: Limiting distribution for the maximal standardized increment of a random walk. Stochastic Process. Appl. 124(9), 2824–2867 (2014). doi: 10.1016/ ISSN 0304-4149MathSciNetCrossRefzbMATHGoogle Scholar
  29. Kabluchko, Z., Schlather, M., de Haan, L.: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37, 2042–2065 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Leadbetter, M.R.: Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65, 291–306 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes, vol. 11. Springer (1983)Google Scholar
  32. Marcus, M.B.: Upper bounds for the asymptotic maxima of continuous Gaussian processes. Ann. Math. Statist. 43, 522–533 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Michna, Z.: Remarks on Pickands Constant. arXiv:0904.3832 (2009)
  34. Molchanov, I., Stucki, K.: Stationarity of multivariate particle systems. Stochastic Process. Appl. 123(6), 2272–2285 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Molchanov, I., Schmutz, M., Stucki, K.: Invariance properties of random vectors and stochastic processes based on the Zonoid concept. Bernoulli 20(3), 1210–1233 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Oesting, M., Kabluchko, Z., Schlather, M.: Simulation of Brown-Resnick processes. Extremes 15(1), 89–107 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Pickands, J. III: Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73 (1969a). ISSN 0002-9947MathSciNetCrossRefzbMATHGoogle Scholar
  38. Pickands, J. III: Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145, 75–86 (1969b)MathSciNetzbMATHGoogle Scholar
  39. Piterbarg, V.I.: Discrete and continuous time extremes of gaussian processes. Extremes 7(2), 161–177 (2005). 2004. ISSN 1386-1999MathSciNetCrossRefzbMATHGoogle Scholar
  40. Piterbarg, V.I.: Twenty Lectures About Gaussian Processes. Atlantic Financial Press, London, New York (2015)zbMATHGoogle Scholar
  41. Samorodnitsky, G.: Probability tails of Gaussian extrema. Stochastic Process. Appl. 38(1), 55–84 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Schlather, M.: Models for stationary max-stable random fields. Extremes 5, 33–44 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  43. Shao, Q.M.: Bounds and estimators of a basic constant in extreme value theory of Gaussian processes. Stat. Sin. 6, 245–258 (1996)MathSciNetzbMATHGoogle Scholar
  44. Stoev, S.A.: On the ergodicity and mixing of max-stable processes. Stochastic Process. Appl. 118, 1679–1705 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Stoev, S.A.: Max-stable processes: representations, ergodic properties and statistical applications. In: Doukhan, P., Lang, G., Surgailis, D., Teyssiere, G. (eds.) Dependence in Probability and Statistics, Lecture Notes in Statistics 200, vol. 200, pp 21–42 (2010)Google Scholar
  46. Strokorb, K., Ballani, F., Schlather, M.: Tail correlation functions of max-stable processes: construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes 18(2), 241–271 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  47. Wang, Y.: Extremes of q-Ornstein-Uhlenbeck Processes. arXiv:1609.00338 (2016)
  48. Wang, Y., Stoev, S.A.: On the structure and representations of max-stable processes. Adv. Appl. Probab. 42(3), 855–877 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. Willekens, E.: On the supremum of an infinitely divisible process. Stochastic Process. Appl. 26, 173–175 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Krzysztof Dȩbicki
    • 1
    Email author
  • Sebastian Engelke
    • 2
  • Enkelejd Hashorva
    • 3
  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.University of LausanneLausanneSwitzerland

Personalised recommendations