Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T05:42:02.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of Maxima of Strongly Dependent Gaussian Processes

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Zhongquan Tan ,
Enkelejd Hashorva  and
Zuoxiang Peng
Zhongquan Tan*
Affiliation:
Soochow University
Enkelejd Hashorva*
Affiliation:
University of Lausanne
Zuoxiang Peng*
Affiliation:
Southwest University
*
∗ Current address: College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, P. R. China.
∗∗ Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland. Email address: [email protected]
∗∗∗ Postal address: School of Mathematics and Statistics, Southwest University, Chongqing 400715, P. R. China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {Xn(t), t∈[0,∞)}, n∈ℕ, be standard stationary Gaussian processes. The limit distribution of t∈[0,T(n)]|Xn(t)| is established as rn(t), the correlation function of {Xn(t), t∈[0,∞)}, n∈ℕ, which satisfies the local and long-range strong dependence conditions, extending the results obtained in Seleznjev (1991).

Keywords

Stationary Gaussian processstrong dependenceBerman's conditionlimit theoremPickands' constant

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 1106 - 1118
Copyright
© Applied Probability Trust 

References

Adler, R. J. (1990). An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes (Inst. Math. Statist. Lecture Notes Monogr. Ser. 12). Institute of Mathematical Statistics, Hayward, CA.CrossRefGoogle Scholar
Albin, J. M. P. and Choi, H. (2010). A new proof of an old result by Pickands. Electron. Commun. Prob. 15, 339–345.CrossRefGoogle Scholar
Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole, Pacific Grove, CA.CrossRefGoogle Scholar
Dȩbicki, K. and Kisowski, P. (2009). A note on upper estimates for Pickands constants. Statist. Prob. Lett. 78, 2046–2051.CrossRefGoogle Scholar
Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207–248.CrossRefGoogle Scholar
Durrett, R. (2004). Probability Theory and Examples. Duxbury Press, Boston.Google Scholar
Ho, H.-C. and McCormick, W. P. (1999). Asymptotic distribution of sum and maximum for Gaussian processes. J. Appl. Prob. 36, 1031–1044.CrossRefGoogle Scholar
Hüsler, J. (1999). Extremes of Gaussian processes, on results of Piterbarg and Seleznjev. Statist. Prob. Lett. 44, 251–258.CrossRefGoogle Scholar
Hüsler, J., Piterbarg, V. and Seleznjev, O. (2003). On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Prob. 13, 1615–1653.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.CrossRefGoogle Scholar
Mittal, Y. and Ylvisaker, D. (1975). Limit distributions for the maximum of stationary Gaussian processes. Stoch. Process. Appl. 3, 1–18.CrossRefGoogle Scholar
Pickands, J. III. (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145, 75–86.Google Scholar
Piterbarg, V. I. (1972). On the paper by J. Pickands ‘Upcrossing probabilities for stationary Gaussian processes’. Vestnik Moscow. Univ. Ser. I Mat. Meh. 27, 25–30 (in Russian). English translation: Moscow Univ. Math. Bull. Google Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence, RI.Google Scholar
Seleznjev, O. V. (1991). Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes. J. Appl. Prob. 28, 17–32.CrossRefGoogle Scholar
Seleznjev, O. (1996). Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments. Adv. Appl. Prob. 28, 481–499.CrossRefGoogle Scholar
Seleznjev, O. (2006). Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields. Extremes 8, 161–169.CrossRefGoogle Scholar
Shao, Q. M. (1996). Bounds and estimators of a basic constant in extreme value theory of Gaussian processes. Statistica Sinica 6, 245–257.Google Scholar
Stamatovic, B. and Stamatovic, S. (2010). Cox limit theorem for large excursions of a norm of Gaussian vector process. Statist. Prob. Lett. 80, 1479–1485.CrossRefGoogle Scholar