IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v160y2023icp318-350.html
   My bibliography  Save this article

Asymptotic behaviour of critical decomposable 2-type Galton–Watson processes with immigration

Author

Listed:
  • Barczy, Mátyás
  • Bezdány, Dániel
  • Pap, Gyula

Abstract

In this paper the asymptotic behaviour of a critical 2-type Galton–Watson process with immigration is described when its offspring mean matrix is reducible, in other words, when the process is decomposable. It is proved that, under second or fourth order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from a critical decomposable 2-type Galton–Watson process with immigration converges weakly. The limit process can be described using one or two independent squared Bessel processes and possibly the unique stationary distribution of an appropriate single-type subcritical Galton–Watson process with immigration. Our results complete and extend the results of Foster and Ney (1978) for some strongly critical decomposable 2-type Galton–Watson processes with immigration.

Suggested Citation

  • Barczy, Mátyás & Bezdány, Dániel & Pap, Gyula, 2023. "Asymptotic behaviour of critical decomposable 2-type Galton–Watson processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 318-350.
    • Handle: RePEc:eee:spapps:v:160:y:2023:i:c:p:318-350
    DOI: 10.1016/j.spa.2023.03.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414923000480
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2023.03.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kevei, Péter & Wiandt, Péter, 2021. "Moments of the stationary distribution of subcritical multitype Galton–Watson processes with immigration," Statistics & Probability Letters, Elsevier, vol. 173(C).
    2. Barczy, M. & Ispány, M. & Pap, G., 2011. "Asymptotic behavior of unstable INAR(p) processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 583-608, March.
    3. Wei, C. Z. & Winnicki, J., 1989. "Some asymptotic results for the branching process with immigration," Stochastic Processes and their Applications, Elsevier, vol. 31(2), pages 261-282, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kristóf Körmendi & Gyula Pap, 2018. "Statistical inference of 2-type critical Galton–Watson processes with immigration," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 169-190, April.
    2. Nikolaos Limnios & Elena Yarovaya, 2020. "Diffusion Approximation of Branching Processes in Semi-Markov Environment," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1583-1590, December.
    3. Robert Jung & Gerd Ronning & A. Tremayne, 2005. "Estimation in conditional first order autoregression with discrete support," Statistical Papers, Springer, vol. 46(2), pages 195-224, April.
    4. Mátyás Barczy & Kristóf Körmendi & Gyula Pap, 2016. "Statistical inference for critical continuous state and continuous time branching processes with immigration," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(7), pages 789-816, October.
    5. Yong-Hua Mao & Yan-Hong Song, 2022. "Criteria for Geometric and Algebraic Transience for Discrete-Time Markov Chains," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1974-2008, September.
    6. Dieter Fiems & Koen De Turck, 2023. "Analysis of Discrete-Time Queues with Branching Arrivals," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
    7. Datta, Somnath & Sriram, T. N., 1995. "A modified bootstrap for branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 275-294, April.
    8. Mátyás Barczy & Márton Ispány & Gyula Pap, 2014. "Asymptotic Behavior of Conditional Least Squares Estimators for Unstable Integer-valued Autoregressive Models of Order 2," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 866-892, December.
    9. Huang, Jianhui & Ma, Chunhua & Zhu, Cai, 2011. "Estimation for discretely observed continuous state branching processes with immigration," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1104-1111, August.
    10. I. Rahimov, 2009. "Asymptotic distributions for weighted estimators of the offspring mean in a branching process," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 568-583, November.
    11. Rahimov, I., 2011. "Estimation of the offspring mean in a supercritical branching process with non-stationary immigration," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 907-914, August.
    12. Xu, Wei, 2014. "Parameter estimation in two-type continuous-state branching processes with immigration," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 124-134.
    13. Qi, Yongcheng & Reeves, Jaxk, 0. "On sequential estimation for branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 41-51, July.
    14. Barczy, M. & Ispány, M. & Pap, G., 2011. "Asymptotic behavior of unstable INAR(p) processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 583-608, March.
    15. Barczy, Mátyás & Körmendi, Kristóf & Pap, Gyula, 2015. "Statistical inference for 2-type doubly symmetric critical irreducible continuous state and continuous time branching processes with immigration," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 92-123.
    16. Isabel Silva & M. Eduarda Silva & Isabel Pereira & Nélia Silva, 2005. "Replicated INAR(1) Processes," Methodology and Computing in Applied Probability, Springer, vol. 7(4), pages 517-542, December.
    17. Shete, Sanjay & Sriram, T. N., 1998. "Fixed precision estimator of the offspring mean in branching processes," Stochastic Processes and their Applications, Elsevier, vol. 77(1), pages 17-33, September.
    18. Sriram, T.N. & Bhattacharya, A. & González, M. & Martínez, R. & del Puerto, I., 2007. "Estimation of the offspring mean in a controlled branching process with a random control function," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 928-946, July.
    19. Rahimov, I., 2008. "Asymptotic distribution of the CLSE in a critical process with immigration," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1892-1908, October.
    20. Li, Zenghu & Ma, Chunhua, 2015. "Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3196-3233.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:160:y:2023:i:c:p:318-350. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.