Advertisement

Extremes

, Volume 16, Issue 2, pp 241–254 | Cite as

Limit theorems for extremes of strongly dependent cyclo-stationary χ-processes

  • Zhongquan Tan
  • Enkelejd Hashorva
Article

Abstract

In this paper, with motivation from the paper of Konstant et al. (Lith Math J 44:196-208, 2004) we derive limit theorems for the maximum of strongly dependent cyclo-stationary \(\chi \)-processes. Further, under a global Hölder condition we show that Seleznjev pth-mean convergence theorem holds.

Keywords

Gaussian process Cyclo-stationary process \(\chi\)-process Gumbel limit law Limit theorem Seleznjev pth-mean convergence theorem Piterbarg inequality 

AMS 2000 Subject Classifications

Primary—60F05; Secondary—60G15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albin, J.P.M.: On extremal theory for stationary processes. Ann. Probab. 18, 92–128 (1990)MathSciNetMATHCrossRefGoogle Scholar
  2. Aronowich, M., Adler, R.J.: Behaviour of \(\chi ^{2}\) processes at extrema. Adv. Appl. Probab. 17, 280–297 (1985)MathSciNetMATHCrossRefGoogle Scholar
  3. Aronowich, M., Adler, R.J.: Extrema and level crossings of \(\chi ^{2}\) processes. Adv. Appl. Probab. 18, 901–920 (1986)MathSciNetMATHCrossRefGoogle Scholar
  4. Azaïs, J.M., Mercadier, C.: Asymptotic Poisson character of extremes in non-stationary Gaussian models. Extremes 6, 301–318 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. Berman, S.M.: Sojourns and extremes of Gaussian processes. Ann. Probab. 2, 999–1026 (1974)MATHCrossRefGoogle Scholar
  6. Berman, S.: Sojourns and extremes of stationary processes. Ann. Probab. 10, 1–46 (1982)MathSciNetMATHCrossRefGoogle Scholar
  7. Berman, M.S.: Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole, Stamford (1992)Google Scholar
  8. Falk, M., Hüsler, J., Reiss, R.-D.: Laws of Small Numbers: Extremes and Rare Events. DMV Seminar Vol. 23, 2nd edn. Birkhäuser, Basel (2010)Google Scholar
  9. Gardner, W.A., Napolitano, A., Paura, L.: Cyclostationarity: Half a century of research. Signal Process. (Elsevier) 86, 639–697 (2006)MATHCrossRefGoogle Scholar
  10. Hüsler, J.: Extreme values and high boundary crossings for locally stationary Gaussian processes. Ann. Probab. 18, 1141–1158 (1990)MathSciNetMATHCrossRefGoogle Scholar
  11. Konstant, D., Piterbarg, V.I.: Extreme values of the cyclostationary Gaussian random process. J. Appl. Probab. 30, 82–97 (1993)MathSciNetMATHCrossRefGoogle Scholar
  12. Konstant, D., Piterbarg, V.I., Stamatovic, S.: Limit theorems for cyclo-stationary \(\chi \)-processes. Lith. Math. J. 44, 196–208 (2004)Google Scholar
  13. Leadbetter, M.R., Rootzén, H.: Extremal theory for stochastic processes. Ann. Appl. Probab. 16, 431–478 (1988)MATHCrossRefGoogle Scholar
  14. Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, Berlin, New York (1983)MATHCrossRefGoogle Scholar
  15. Lindgren, G.: Extreme values and crossings for the \(\chi ^{2}\)-process and other functions of multidimensional Gaussian proceses with reliability applications. Adv. Appl. Probab. 12, 746–774 (1980)MathSciNetMATHCrossRefGoogle Scholar
  16. Lindgren, G.: Slepian models for \(\chi ^{2}\)-process with dependent components with application to envelope upcrossings. J. Appl. Probab. 26, 36–49 (1989)MathSciNetMATHCrossRefGoogle Scholar
  17. Piterbarg, V.I.: High excursions for nonstationary generalized chi-square processes. Stoch. Proc. Appl. 53, 307–337 (1994)MathSciNetMATHCrossRefGoogle Scholar
  18. Piterbarg, V.I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields. In: Transl. Math. Monographs, Vol. 148. AMS, Providence, RI (1996)Google Scholar
  19. Piterbarg, V.I.: Large deviations of a storage process with fractional Browanian motion as input. Extremes 4, 147–164 (2001)MathSciNetMATHCrossRefGoogle Scholar
  20. Piterbarg, V.I., Stamatovic, S.: Limit theorem for high level \(\alpha \)-upcrossings by \(\chi \)-process. Theory Probab. Appl. 48, 734–741 (2004)MathSciNetCrossRefGoogle Scholar
  21. Seleznjev, O.: Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields. Extremes 8, 161–169 (2006)MathSciNetCrossRefGoogle Scholar
  22. Sharpe, K.: Some properties of the crossing process generated by a stationary \(\chi ^{2}\)-process. Adv. Appl. Probab. 10, 373–391 (1978)MathSciNetMATHCrossRefGoogle Scholar
  23. Stamatovic, B., Stamatovic, S.: Cox limit theorem for large excursions of a norm of Gaussian vector process. Stat. Prob. Lett. 80, 1479–1485 (2010)MathSciNetMATHCrossRefGoogle Scholar
  24. Tan, Z., Hashorva, E.: Exact asymptotics and limit theorems for supremum of stationary \(\chi \)-processes over a random interval. Preprint (2012)Google Scholar
  25. Tan, Z., Hashorva, E., Peng, Z.: Asymptotics of maxima of strongly dependent Gaussian processes. J. Appl. Probab. 49, 1102–1118 (2012) (to appear)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.College of Mathematics, Physics and Information EngineeringJiaxing UniversityJiaxingPeople’s Republic of China
  2. 2.Department of Actuarial Science, Faculty of Business and EconomicsUniversity of LausanneLausanneSwitzerland

Personalised recommendations