×

On asymptotic constants in the theory of extremes for Gaussian processes. (English) Zbl 1298.60043

Summary: 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.

MSC:

60G15 Gaussian processes
60G70 Extreme value theory; extremal stochastic processes
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[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. · Zbl 0747.60039
[2] Adler, R.J. and Taylor, J.E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics . New York: Springer. · Zbl 1149.60003
[3] Albin, J.M.P. and Choi, H. (2010). A new proof of an old result by Pickands. Electron. Commun. Probab. 15 339-345. · Zbl 1227.60068
[4] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77 . New York: Springer. · Zbl 0679.60013
[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. · Zbl 0853.65147
[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. · Zbl 1079.60031
[7] Azaïs, J.M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields . Hoboken, NJ: Wiley. · Zbl 1168.60002
[8] Bender, C. and Parczewski, P. (2010). Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus. Bernoulli 16 389-417. · Zbl 1248.60044
[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. · Zbl 0809.60046
[10] Bickel, P.J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095. · Zbl 0275.62033
[11] Billingsley, P. (1968). Convergence of Probability Measures . New York: Wiley. · Zbl 0172.21201
[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. · Zbl 1013.60030
[14] Davies, R.B. and Harte, D.S. (1987). Tests for Hurst effect. Biometrika 74 95-101. · Zbl 0612.62123
[15] De\ogonek bicki, K. (2002). Ruin probability for Gaussian integrated processes. Stochastic Process. Appl. 98 151-174. · Zbl 1059.60047
[16] De\ogonek bicki, K. (2006). Some properties of generalized Pickands constants. Theory Probab. Appl. 50 290-298. · Zbl 1089.60035
[17] De\ogonek bicki, K. and Kisowski, P. (2008). A note on upper estimates for Pickands constants. Statist. Probab. Lett. 78 2046-2051. · Zbl 1283.60068
[18] De\ogonek bicki, K. and Mandjes, M. (2011). Open problems in Gaussian fluid queueing theory. Queueing Syst. 68 267-273. · Zbl 1275.60039
[19] Dieker, A.B. (2005). Conditional limit theorem for queues with Gaussian input, a weak convergence approach. Stochastic Process. Appl. 115 849-873. · Zbl 1073.60085
[20] Dieker, T. (2002). Simulation of fractional Brownian motion. Master’s thesis. Amsterdam, Vrije Universiteit.
[21] De\ogonek bicki, K., Michna, Z. and Rolski, T. (2003). Simulation of the asymptotic constant in some fluid models. Stoch. Models 19 407-423. · Zbl 1039.60040
[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. · Zbl 1268.60075
[23] Hüsler, J. (1999). Extremes of a Gaussian process and the constant \(H_{\alpha}\). Extremes 2 59-70. · Zbl 0944.60040
[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. · Zbl 1038.60040
[25] Kobel’kov, S.G. (2005). On the ruin problem for a Gaussian stationary process. Theory Probab. Appl. 49 155-163. · Zbl 1089.60022
[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 . 1202.4970
[28] Michna, Z. (1999). On tail probabilities and first passage times for fractional Brownian motion. Math. Methods Oper. Res. 49 335-354. · Zbl 0953.60016
[29] Pickands, J. III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145 75-86. · Zbl 0206.18901
[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. · Zbl 0841.60024
[31] Seleznjev, O. (1996). Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments. Adv. in Appl. Probab. 28 481-499. · Zbl 0861.60033
[32] Shao, Q.M. (1996). Bounds and estimators of a basic constant in extreme value theory of Gaussian processes. Statist. Sinica 6 245-257. · Zbl 0841.60036
[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. · Zbl 1200.62090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.