Advertisement

Extremes

, Volume 21, Issue 4, pp 551–579 | Cite as

The tail process revisited

  • Hrvoje Planinić
  • Philippe Soulier
Article

Abstract

The tail measure of a regularly varying stationary time series has been recently introduced. It is used in this contribution to reconsider certain properties of the tail process and establish new ones. A new formulation of the time change formula is used to establish identities, some of which were indirectly known and some of which are new.

Keywords

Regularly varying time series Tail process Tail measure 

AMS 2000 Subject Classifications

60G70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

Section 5 owes a lot to Enkelejd Hashorva who brought the references Hashorva (2016) and Dȩbicki and Hashorva (2017) to our attention as well as the formula (3.20). The research of the first author is supported in part by Croatian Science Foundation under the project 3526. The research of the second author is partially supported by LABEX MME-DII.

References

  1. Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119(4), 1055–1080 (2009)MathSciNetCrossRefGoogle Scholar
  2. Basrak, B., Tafro, A.: A complete convergence theorem for stationary regularly varying multivariate time series. Extremes 19, 549–560 (2016)MathSciNetCrossRefGoogle Scholar
  3. Basrak, B., Krizmanić, D., Segers, J.: A functional limit theorem for dependent sequences with infinite variance stable limits. Ann. Probab. 40(5), 2008–2033 (2012)MathSciNetCrossRefGoogle Scholar
  4. Basrak, B., Planinić, H., Soulier, P.: An invariance principle for sums and record times of regularly varying stationary sequences. arXiv:1609.00687 (2016)
  5. Davis, R.A., Hsing, T.: Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23(2), 879–917 (1995)MathSciNetCrossRefGoogle Scholar
  6. Dȩbicki, K., Hashorva, E.: On extremal index of max-stable processes. Probab. Math. Stat. 37, 299–317 (2017)MathSciNetzbMATHGoogle Scholar
  7. Dombry, C., Kabluchko, Z.: Ergodic decompositions of stationary max-stable processes in terms of their spectral functions. arXiv:1601.00792(2016)
  8. Dombry, C., Hahorva, E., Soulier, P.: Tail measure and tail spectral process of regularly varying time series. arXiv:1710.08358 (2017)
  9. Guivarc’h, Y., Le Page, E.: Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 52(2), 503–574 (2016)MathSciNetCrossRefGoogle Scholar
  10. Hashorva, E.: Representations of max-stable processes via exponential tilting. arXiv:1605.03208 (2016)
  11. Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94), 121–140 (2006)MathSciNetCrossRefGoogle Scholar
  12. Janßen, A.: Spectral tail processes and max-stable approximations of multivariate regularly varying time series. arXiv e-prints (2017)Google Scholar
  13. Kallenberg, O.: Random Measures, Theory and Applications, vol. 77 of Probability Theory and Stochastic Modelling. Springer, New York (2017)CrossRefGoogle Scholar
  14. Mikosch, T., Wintenberger, O.: The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains. Probab. Theory Relat. Fields 159(1-2), 157–196 (2014)MathSciNetCrossRefGoogle Scholar
  15. Mikosch, T., Wintenberger, O.: A large deviations approach to limit theorem for heavy-tailed time series. Probab. Theory Relat. Fields 166(1-2), 233–269 (2016)CrossRefGoogle Scholar
  16. Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Applied Probability, vol. 4. Springer, New York (1987)CrossRefGoogle Scholar
  17. Samorodnitsky, G., Owada, T.: Tail measures of stochastic processes or random fields with regularly varying tails. Preprint (2012)Google Scholar
  18. Segers, J., Zhao, Y., Meinguet, T.: Polar decomposition of regularly varying time series in star-shaped metric spaces. Extremes 20(3), 539–566 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia
  2. 2.Département de MathématiquesUniversité Paris NanterreNanterreFrance

Personalised recommendations