Skip to main content
SpringerLink
Log in

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

  • Published:
Siberian Advances in Mathematics Aims and scope Submit manuscript

Abstract

We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

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.

Similar content being viewed by others

References

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

    Book  MATH  Google Scholar 

  2. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation (Cambridge Univ. Press, Cambridge, 1989).

    MATH  Google Scholar 

  3. A. A. Borovkov, Probability Theory (Izd. IM SB RAS, Novosibirsk, 2009). [Probability Theory (Springer, London, 2013)].

    Google Scholar 

  4. K. B. Erickson, “Strong renewal theorems with infinite mean,” Trans. Amer.Math. Soc. 151, 263 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Feller, An introduction to Probability Theory and its Applications. Vol. 2 (John Wiley and Sons, New York, 1971).

    MATH  Google Scholar 

  6. F. R. Gantmacher, The Theory ofMatrices (Nauka, Moscow, 1967) [The Theory ofMatrices (AMSChelsea Publishing, Providence, RI, 1998)].

    Google Scholar 

  7. V. A. Il’in and E. G. Poznyak, Linear Algebra (Fizmatlit, Moscow, 2001) [Linear Algebra (Mir, Moscow, 1986)].

    MATH  Google Scholar 

  8. B. A. Rogozin and M. S. Sgibnev, “Banach algebras of absolutely continuous measures on the straight line,” Sib. Matem. Zh. 20, 119 (1979) [SiberianMath. J. 20, 86 (1979)].

    MATH  Google Scholar 

  9. B. A. Sevast’yanov, Branching Processes (Nauka, Moscow, 1971) [in Russian].

    MATH  Google Scholar 

  10. M. S. Sgibnev, “Banach algebras of measures of class G(γ),” Sib. Matem. Zh. 29, no 4, 162 (1979) [Siberian Math. J. 20, 647 (1988)].

    Google Scholar 

  11. V. M. Shurenkov, “A note on a multi-dimensional renewal equation,” Teor. Veroyatn. Primen. 20, 848 (1975) [Theory Probab. Appl. 20, 833 (1976)].

    MATH  Google Scholar 

  12. V. Topchiĭ, “Renewal measure density for distributions with regularly varying tails of order α ∈ (0, 1/2],” in Workshop on Branching Processes and Their Applications, 109 (Springer, Heidelberg–Dordrecht–London–New York, 2010).

    Google Scholar 

  13. V. A. Topchiĭ, “Derivative of renewal density with infinite moment with α ∈ (0, 1/2],” Sib. Elektron.Mat. Izv. 7, 340 (2010) [in Russian].

    MATH  Google Scholar 

  14. V. A. Topchiĭ, “The asymptotic behaviour of derivatives of the renewal function for distributions with infinite first moment and regularly varying tails of index β ∈ (1/2, 1],” Diskretn. Mat. 24, no. 2, 123 (2012) [Discrete Math. Appl. 22, 315 (2012)].

    Article  MATH  Google Scholar 

  15. V. A. Topchiĭ, “‘Two-dimensional renewal theorems with weak moment conditions and critical Bellman–Harris branching processes,” Diskretn. Mat. 27, no. 1, 123 (2015) [Discrete Math. Appl. 26, 51 (2016)].

    Article  Google Scholar 

  16. V. A. Vatutin, “Discrete limit distributions of the number of particles in a Bellman–Harris branching process with several types of particles,” Teor. Veroyatn. Primen. 24, 503 (1979) [Theory Probab. Appl. 24, 509 (1980)].

    MATH  Google Scholar 

  17. V. A. Vatutin, “On a class of critical Bellman–Harris branching processes with several types of particles,” Teor. Veroyatn. Primen. 25, 771 (1980) [Theory Probab. Appl. 25, 760 (1981)].

    MathSciNet  MATH  Google Scholar 

  18. V. Vatutin, A. Iksanov, and V. Topchiĭ, “A two-type Bellman–Harris process initiated by a large number of particles,” Acta Appl.Math. 138, 279 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. A. Vatutin and V. A. Topchiĭ, “Catalytic branching random walks in Zd with branching at the origin,” Mat. Trudy 14, no. 2, 28 (2011) [Siberian Adv.Math. 23, 103 (2013)].

    MATH  Google Scholar 

  20. V. A. Vatutin and V. A. Topchiĭ, “Critical Bellman–Harris branching processes with long-living particles,” Trudy Mat. Inst. Steklov 282, 257 (2013) [Proc. Steklov Inst. Math. 282, 243 (2013)].

    MathSciNet  MATH  Google Scholar 

  21. V. A. Vatutin and V. A. Topchiĭ, “A key renewal theorem for heavy tail distributions with β ∈ (0, 0.5],” Teor. Veroyatn. Primen. 58, 387 (2013) [Theory Probab. Appl. 58, 333 (2014)].

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Omsk Division, Omsk, 644099, Russia

    V. A. Topchiĭ

Authors
  1. V. A. Topchiĭ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Topchiĭ.

Additional information

Original Russian Text © V.A. Topchiĭ, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 139–192.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Topchiĭ, V.A. On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders. Sib. Adv. Math. 28, 115–153 (2018). https://doi.org/10.3103/S1055134418020037

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1055134418020037

Keywords

Search

Navigation