• Bernoulli
  • Volume 20, Number 3 (2014), 1600-1619.

On asymptotic constants in the theory of extremes for Gaussian processes

A.B. Dieker and B. Yakir

Full-text: Open access


This paper gives a new representation of Pickands’ constants, which arise in the study of extremes for a variety of Gaussian processes. Using this representation, we resolve the long-standing problem of devising a reliable algorithm for estimating these constants. A detailed error analysis illustrates the strength of our approach.

Article information

Bernoulli, Volume 20, Number 3 (2014), 1600-1619.

First available in Project Euclid: 11 June 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

extremes Gaussian processes Monte Carlo simulation Pickands’ constants


Dieker, A.B.; Yakir, B. On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20 (2014), no. 3, 1600--1619. doi:10.3150/13-BEJ534.

Export citation


  • [1] Adler, R.J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Institute of Mathematical Statistics Lecture Notes—Monograph Series 12. Hayward, CA: IMS.
  • [2] Adler, R.J. and Taylor, J.E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. New York: Springer.
  • [3] Albin, J.M.P. and Choi, H. (2010). A new proof of an old result by Pickands. Electron. Commun. Probab. 15 339–345.
  • [4] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77. New York: Springer.
  • [5] Asmussen, S., Glynn, P. and Pitman, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Probab. 5 875–896.
  • [6] Azaïs, J.M. and Wschebor, M. (2005). On the distribution of the maximum of a Gaussian field with $d$ parameters. Ann. Appl. Probab. 15 254–278.
  • [7] Azaïs, J.M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Hoboken, NJ: Wiley.
  • [8] Bender, C. and Parczewski, P. (2010). Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus. Bernoulli 16 389–417.
  • [9] Berman, S.M. (1992). Sojourns and Extremes of Stochastic Processes. The Wadsworth & Brooks/Cole Statistics/Probability Series. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software.
  • [10] Bickel, P.J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • [11] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
  • [12] Bogachev, V.I. (1998). Gaussian Measures. Mathematical Surveys and Monographs 62. Providence, RI: Amer. Math. Soc.
  • [13] Burnecki, K. and Michna, Z. (2002). Simulation of Pickands constants. Probab. Math. Statist. 22 193–199.
  • [14] Davies, R.B. and Harte, D.S. (1987). Tests for Hurst effect. Biometrika 74 95–101.
  • [15] Dȩbicki, K. (2002). Ruin probability for Gaussian integrated processes. Stochastic Process. Appl. 98 151–174.
  • [16] Dȩbicki, K. (2006). Some properties of generalized Pickands constants. Theory Probab. Appl. 50 290–298.
  • [17] Dȩbicki, K. and Kisowski, P. (2008). A note on upper estimates for Pickands constants. Statist. Probab. Lett. 78 2046–2051.
  • [18] Dȩbicki, K. and Mandjes, M. (2011). Open problems in Gaussian fluid queueing theory. Queueing Syst. 68 267–273.
  • [19] Dieker, A.B. (2005). Conditional limit theorem for queues with Gaussian input, a weak convergence approach. Stochastic Process. Appl. 115 849–873.
  • [20] Dieker, T. (2002). Simulation of fractional Brownian motion. Master’s thesis. Amsterdam, Vrije Universiteit.
  • [21] Dȩbicki, K., Michna, Z. and Rolski, T. (2003). Simulation of the asymptotic constant in some fluid models. Stoch. Models 19 407–423.
  • [22] Harper, A.J. (2013). Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab. 23 584–616.
  • [23] Hüsler, J. (1999). Extremes of a Gaussian process and the constant $H_{\alpha}$. Extremes 2 59–70.
  • [24] Hüsler, J., Piterbarg, V. and Seleznjev, O. (2003). On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 1615–1653.
  • [25] Kobel’kov, S.G. (2005). On the ruin problem for a Gaussian stationary process. Theory Probab. Appl. 49 155–163.
  • [26] Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. New York: Springer.
  • [27] Meka, R. (2010). A PTAS for computing the supremum of Gaussian processes. Available at arXiv:1202.4970.
  • [28] Michna, Z. (1999). On tail probabilities and first passage times for fractional Brownian motion. Math. Methods Oper. Res. 49 335–354.
  • [29] Pickands, J. III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145 75–86.
  • [30] Piterbarg, V.I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148. Providence, RI: Amer. Math. Soc.
  • [31] Seleznjev, O. (1996). Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments. Adv. in Appl. Probab. 28 481–499.
  • [32] Shao, Q.M. (1996). Bounds and estimators of a basic constant in extreme value theory of Gaussian processes. Statist. Sinica 6 245–257.
  • [33] Siegmund, D., Yakir, B. and Zhang, N. (2010). Tail approximations for maxima of random fields by likelihood ratio transformations. Sequential Anal. 29 245–262.