SpringerLink
Log in

Maxima and sums of non-stationary random length sequences

  • Published:
Extremes Aims and scope Submit manuscript

Abstract

We study non-stationary random length sequences of random variables with regularly varying tails. Tail and extremal indexes of their maxima and linear combinations are found. We obtain conditions when both sums and maxima of these sequences have the same tail and extremal indexes. Their extremal index corresponds to the tail index of the most heavy-tailed random variable in the sequence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ancona-Navarrete, M.A., Tawn, J.A.: A comparison of Methods for Estimating the Extremal Index. Extremes 3:1, 5–38 (2000)

    Article  MathSciNet  Google Scholar 

  • Asmussen, S., Foss, S.: Regular variation in a fixed-point problem for single- and multi-class branching processes and queues. Branching Processes and Applied Probability. Papers in Honour of Peter Jagers. Adv. Appl. Prob. 50A, 47–61 (2018)

    Article  Google Scholar 

  • Baldi, P., Frasconi, P., Smyth, P.: Modeling the Internet and the Web: Probabilistic Methods and Algorithms. Wiley, New York (2003)

  • Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080 (2009)

    Article  MathSciNet  Google Scholar 

  • Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge university press, Cambridge (1987)

    Book  Google Scholar 

  • Chen, N., Litvak, N., Olvera-Cravioto, M.: Pagerank in Scale-Free Random Graphs. WAW 2014, LNCS 8882. In: Bonato, A. (ed.) , pp 120–131. Springer, Switzerland (2014)

  • de Haan, L.: Convergence of heteroscedastic extremes. Stat. Probab. Lett. 101, 38–39 (2015)

    Article  MathSciNet  Google Scholar 

  • Faÿ, G., González-Arévalo, B., Mikosch, T., Samorodnitsky, G.: Modeling teletraffic arrivals by a Poisson cluster process. Queueing Syst. 54(2), 121–140 (2006)

    Article  MathSciNet  Google Scholar 

  • Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. R. Statist. Soc. B. 65, 545–556 (2003)

    Article  MathSciNet  Google Scholar 

  • Goldaeva, A.A.: Indices of multivariate recurrent stochastic sequences. In: Shiryaev, A.N. (ed.) Modern problem of mathematics and mechanics VIII(3). (in Russian), pp 42–51. Moscow State University ISBN 978–5–211–05652–7 (2013)

  • Goldaeva, A.A., Lebedev, A.V.: On extremal indices greater than one for a scheme of series. Lith. Math. J. 58(4), 384–398 (2018). https://doi.org/10.1007/s10986-018-9407-2

    Article  MathSciNet  Google Scholar 

  • Hult, H., Samorodnitsky, G.: Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14(3), 838–864 (2008)

    Article  MathSciNet  Google Scholar 

  • Jelenkovic, P.R., Olvera-Cravioto, M.: Information ranking and power laws on trees. Adv. Appl. Prob. 42(4), 1057–1093 (2010)

    Article  MathSciNet  Google Scholar 

  • Jelenkovic, P.R., Olvera-Cravioto, M.: Maximums on trees. Stoch. Process. Appl. 125, 217–232 (2015)

    Article  MathSciNet  Google Scholar 

  • Jessen, A.H., Mikosch, T.: Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80, 171–192 (2006)

    Article  MathSciNet  Google Scholar 

  • Leadbetter, M.R., Lingren, G., Rootzén, H.: Extremes and Related Properties of Random Sequence and Processes. Ch. 3. Springer, New York (1983)

  • Lebedev, A.V.: Activity maxima in some models of information networks with random weights and heavy tails. Probl. Inf. Transm. 51(1), 66–74 (2015a)

    Article  MathSciNet  Google Scholar 

  • Lebedev, A.V.: Extremal indices in the series scheme and their applications. Informatics Appl. 9(3), 39–54 (2015b). (in Russian)

    Google Scholar 

  • Lee, J., Olvera-Cravioto, M.: Pagerank on inhomogeneous random digraphs. ar**v:1707.02492, pp. 1–57 (2017)

  • Markovich, N.M.: Modeling clusters of extreme values. Extremes 17(1), 97–125 (2014)

    Article  MathSciNet  Google Scholar 

  • Markovich, N.M.: Erratum to: modeling clusters of extreme values. Extremes 19(1), 139–142 (2016)

    Article  MathSciNet  Google Scholar 

  • Markovich, N.M.: Clusters of extremes: modeling and examples. Extremes 20, 519–538 (2017)

    Article  MathSciNet  Google Scholar 

  • Olvera-Cravioto, M.: Asymptotics for weighted random sums. Adv. Appl. Prob. 44(4), 1142–1172 (2012)

    Article  MathSciNet  Google Scholar 

  • Robert, C.Y.: Automatic declustering of rare events. Biometrika 100, 587–606 (2013)

    Article  MathSciNet  Google Scholar 

  • Robert, C.Y., Segers, J.: Tails of random sums of a heavy-tailed number of light-tailed terms. Insur. Math. Econ. 43, 85–92 (2008)

    Article  MathSciNet  Google Scholar 

  • Rootzén, H.: Maxima and exceedances of stationary Markov chains. Adv. Appl. Probab. 20, 371–390 (1988)

    Article  MathSciNet  Google Scholar 

  • Tillier, C., Wintenberger, O.: Regular variation of a random length sequence of random variables and application to risk assessment. Extremes. 21, 27–56 (2018)

    Article  MathSciNet  Google Scholar 

  • Volkovich, Y.V., Litvak, N.: Asymptotic analysis for personalized web search. Adv. Appl. Prob. 42(2), 577–604 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors were partly supported by the Russian Foundation for Basic Research (grant No. 19-01-00090).

Author information

Authors and Affiliations

  1. Russian Academy of Sciences, V. A. Trapeznikov Institute of Control Sciences, Profsoyuznaya 65, 117997, Moscow, Russia

    Natalia M. Markovich & Igor V. Rodionov

Authors
  1. Natalia M. Markovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor V. Rodionov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Natalia M. Markovich.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Markovich, N.M., Rodionov, I.V. Maxima and sums of non-stationary random length sequences. Extremes 23, 451–464 (2020). https://doi.org/10.1007/s10687-020-00372-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10687-020-00372-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation