Skip to main content
SpringerLink
Log in

Branching Processes as Sums of Dependent Random Variables

  • Conference paper
Book cover Branching Processes

Part of the book series: Lecture Notes in Statistics ((LNS,volume 99))

Abstract

It is well-known that branching processes can be represented as a sum of random number of independent random variables. This relation serves as a starting point for investigation of the branching process. However, characteristics of the population representing as sums of dependent terms are not usually amenable to investigation by the traditional methods in the theory of branching processes. If we use some results in the theory of summation of dependent variables:

  1. (a)

    it is possible to study some new characteristics of population in the branching process without immigration (the number of r-tuples at time t1 that have at time t2>t1 nonempty offspring sets differing from each other in number by at most d; the number of pairs of particles having the same number of offspring at time t; reduced branching processes and so on ).

  2. (b)

    it is possible to study branching processes with immigration, depending on reproduction.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Ambrosimov, A.S. (1976) Normal laws in the scheme of sums of dependent random variables considered by B. A. Sevastyanov. Theory Probab. Appl., 21, 183–188.

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen, S., Hering, H. (1976) Strong limit theorems for supercritical migration-branching processes. Math. Scand., 39, 327–342.

    MathSciNet  Google Scholar 

  • Athreya, K. B., Parthasarathy, P. R., Sankaranarayanan, G. (1974) Supercritical age-dependent branching processes with immigration. J. Appl. Probab., 11, 695–702.

    Article  MathSciNet  MATH  Google Scholar 

  • Badalbayev, I., S., Zubkov, A., M. (1983) A limit theorem for the sequence of branching processes with immigration. Theory Probab. Appl., 28, 382–388Theory Probab. Appl., 28, 382-388Theory Probab. Appl., 28, 382-388.

    Google Scholar 

  • Durham, S. D. (1971) A problem concerning generalized age-dependent branching processes with immigration. Ann. Math. Stat., 42, 1121–1123.

    Article  MathSciNet  MATH  Google Scholar 

  • Eagleson, G.K. (1975) Martingale convergence to mixtures of infinitely divisible laws. Ann. Probab., 3, 557–562.

    Article  MathSciNet  MATH  Google Scholar 

  • Foster, J. (1969) Branching processes involving immigration. Ph. D. Thesis, University of Wisconsin.

    Google Scholar 

  • Foster, J., Williamson, J. (1971) Limit theorems for the Galton-Watson process with time dependent immigration. Z. Wahrschein, und Verw. Geb., 20, 227–235.

    Article  MathSciNet  Google Scholar 

  • Nagayev, S. (1975) A limit theorem for branching processes with immigration. Theory Probab. Appl. 20, 178–180.

    Google Scholar 

  • Nagayev, S., Asadullin, M. (1985) On certain scheme for summation of random number of independent random variables with applications to branching processes. Dokl. Akad. Nauk SSSR, 285(2), 293–296.

    MathSciNet  Google Scholar 

  • Rahimov, I. (1986) Critical branching processes with infinite variance and decreasing immigration. Theory Probab. Appl., 31, 88–100.

    Article  Google Scholar 

  • Rahimov, I. (1987) A limit theorem for random sums of dependent indicators and its applications in the theory of branching processes Theory Probab. Appl., 32,2, 290–298.

    Article  Google Scholar 

  • Seneta E. (1970) An explicit-limit theorem for the critical Galton-Watson process with immigration. JRSS 32, 149–152.

    MathSciNet  MATH  Google Scholar 

  • Sevastyanov, B. A. (1957) Limit theorems for a special type branching random processes. Theory Probab. Appl., 2, 339–348.

    MathSciNet  Google Scholar 

  • Sevastyanov, B.A. (1972) Poisson limit law in the scheme of sums of dependent random variables. Theory Probab. Appl., 17, 695–699.

    Article  Google Scholar 

  • Shurenkov, B. M. (1976) Two limit theorems for critical branching processes. Theory Probab. Appl., 21, 548–558.

    Google Scholar 

  • Silverman, B. and Brown, T. (1978) Short distances, flat triangles and Poisson limits. J. Appl. Probab. 15, 815–825.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Romanovski Math. Inst. Tashkent and METU, Ankara, Turkey

    I. Rahimov

  2. 700180, Yunusabad, 15-31-74, Tashkent, Uzbekistan

    I. Rahimov

Authors
  1. I. Rahimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Stochastic Analysis Group, CMA, Australian National University, Canberra, ACT, 0200, Australia

    C. C. Heyde

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag New York, Inc.

About this paper

Cite this paper

Rahimov, I. (1995). Branching Processes as Sums of Dependent Random Variables. In: Heyde, C.C. (eds) Branching Processes. Lecture Notes in Statistics, vol 99. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2558-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2558-4_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-97989-2

  • Online ISBN: 978-1-4612-2558-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation