Skip to main content
SpringerLink
Log in

On the Application of Slowly Varying Functions with Remainder in the Theory of Markov Branching Processes with Mean One and Infinite Variance

  • Published:
Ukrainian Mathematical Journal Aims and scope

We investigate the applications of slowly varying functions (in Karamata’s sense) to the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment but regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine the known limit results.

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

Access this article

Log in via an institution

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

  1. S. Asmussen and H. Hering, “Branching processes,” Progress in Probability and Statistics, 3, Birkhäuser Boston, Inc., Boston, MA (1983).

  2. K. B. Athreya and P. E. Ney, Branching Processes, Springer-Verlag, New York–Heidelberg (1972).

    Book  Google Scholar 

  3. N. H. Bingham, C. M. Goldie, and J. L. Teugels, “Regular variation,” Encyclopedia of Mathematics and Its Applications, 27, Cambridge Univ. Press, Cambridge (1987).

  4. W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2 [in Russian], Mir, Moscow (1967).

  5. T. E. Harris, The Theory of Branching Processes, Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, NJ (1963).

  6. A. A. Imomov, “On conditioned limit structure of the Markov branching process without finite second moment,” Malays. J. Math. Sci., 11, No. 3, 393–422 (2017).

    MathSciNet  MATH  Google Scholar 

  7. A. A. Imomov, “On Markov analogue of Q-processes with continuous time,” Theory Probab. Math. Statist., 84, 57–64 (2012).

    Article  MathSciNet  Google Scholar 

  8. A. N. Kolmogorov and N. A. Dmitriev, “Branching stochastic process,” Dokl. Akad. Nauk SSSR, 56, 5–8 (1947).

    Google Scholar 

  9. A. G. Pakes, “Critical Markov branching process limit theorems allowing infinite variance,” Adv. Appl. Probab., 42, 460–488 (2010).

    Article  MathSciNet  Google Scholar 

  10. E. Seneta, “Regularly varying functions,” Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin–New York (1976).

  11. B. A. Sevastyanov, “Theory of branching stochastic processes,” Usp. Mat. Nauk, (46), No. 6, 47–99 (1951).

  12. V. M. Zolotarev, “More exact statements of several theorems in the theory of branching processes,” Theory Probab. Appl., 2, 245–253 (1957).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Karshi State University, Karshi, Uzbekistan

    A. Imomov & A. Meyliyev

Authors
  1. A. Imomov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Meyliyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Imomov.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 8, pp. 1056–1066, August, 2021. Ukrainian DOI: 10.37863/umzh.v73i8.684.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Imomov, A., Meyliyev, A. On the Application of Slowly Varying Functions with Remainder in the Theory of Markov Branching Processes with Mean One and Infinite Variance. Ukr Math J 73, 1225–1237 (2022). https://doi.org/10.1007/s11253-022-01988-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-022-01988-5

Search

Navigation