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Lithuanian Mathematical Journal

, Volume 58, Issue 4, pp 457–479 | Cite as

Joint functional convergence of partial sums and maxima for linear processes*

  • Danijel KrizmanićEmail author
Article
  • 17 Downloads

Abstract

For linear processes with independent identically distributed innovations that are regularly varying with tail index α ∈ (0, 2), we study the functional convergence of the joint partial-sum and partial-maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes, which enable the functional convergence in the space of ℝ2-valued càdlàg functions on [0, 1] with the Skorokhod weak M2 topology.We also obtain a joint convergence in the M2 topology on the first coordinate and in theM1 topology on the second coordinate.

Keywords

extremal process functional limit theorem linear process regular variation Skorokhod M2 topology stable Lévy process 

MSC

60F17 60G52 

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References

  1. 1.
    J. Astrauskas, Limit theorems for sums of linearly generated random variables, Lith.Math. J., 23(2):127–134, 1983.MathSciNetCrossRefGoogle Scholar
  2. 2.
    F. Avram and M. Taqqu, Weak convergence of sums of moving averages in the α-stable domain of attraction, Ann. Probab., 20(1):483–503, 1992.MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. Balan, A. Jakubowski, and S. Louhichi, Functional convergence of linear processes with heavy-tailed innovations, J. Theor. Probab., 29(2):491–526, 2016.MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Basrak and D. KrizmanićA limit theorem for moving averages in the α-stable domain of attraction, Stochastic Processes Appl., 124(2):1070–1083, 2014.MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Basrak, D. Krizmanić, and J. Segers, A functional limit theorem for partial sums of dependent random variables with infinite variance, Ann. Probab., 40(5):2008–2033, 2012.MathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Basrak and A. Tafro, A complete convergence theorem for stationary regularly varying multivariate time series, Extremes, 19(3):549–560, 2016.MathSciNetCrossRefGoogle Scholar
  7. 7.
    B. Böttcher, EmbeddedMarkov chain approximations in Skorokhod topologies, arXiv:1409.4656.Google Scholar
  8. 8.
    T.L. Chow and J.L. Teugels, The sum and the maximum of i.i.d. random variables, in P. Mandl and M. Hušková (Eds.), Proceedings of the Second Prague Symposium of Asymptotic Statistics, 21–25, August, 1978, North-Holland, Amsterdam, New York, 1979, pp. 81–92.Google Scholar
  9. 9.
    D. Cline, Infinite series of random variables with regularly varying tails, Tecnical Report No. 83-24, Institute of Applied Mathematics and Statistics, University of British Columbia, Canada, 1983.Google Scholar
  10. 10.
    R. Davis and S.I. Resnick, Limit theorems for moving averages with regularly varying tail probabilities, Ann. Probab., 13(1):179–195, 1985.MathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Krizmanić, Weak convergence of partial maxima processes in theM1 topology, Extremes, 17(3):447–465, 2014.MathSciNetCrossRefGoogle Scholar
  12. 12.
    D. Krizmanić, A note on joint functional convergence of partial sum and maxima for linear processes, Stat. Probab. Lett., 138:42–46, 2018.MathSciNetCrossRefGoogle Scholar
  13. 13.
    S.I. Resnick, Point processes, regular variation and weak convergence, Adv. Appl. Probab., 18(1):66–138, 1986.MathSciNetCrossRefGoogle Scholar
  14. 14.
    S.I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987.CrossRefGoogle Scholar
  15. 15.
    S.I. Resnick, Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Springer, New York, 2007.zbMATHGoogle Scholar
  16. 16.
    K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge, 1999.zbMATHGoogle Scholar
  17. 17.
    W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, New York, 2002.zbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RijekaRijekaCroatia

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