Abstract
The life period of a branching process with immigration begins at the moment T and has length τ if the number of particles μ(T −0)=0, μ(t)>0 for all T⩽t<T+τ, and μ(T+τ)=0 (the trajectories of the process are assumed to be continuous from the right). For a critical Markov branching process is obtained a limit theorem on the behavior of μ(t) under the condition that τ>t and T=0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
A. M. Zubkov, “Life periods of a branching process with immigration,” Teor. Veroyatn. Ee Prim.,17, No. 1, 179–188 (1972).
B. A. Sevast'yanov, Branching Processes [in Russian], Nauka, Moscow (1971).
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press (1966).
I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1971).
E. Seneta, “An explicit limit theorem for the critical Galton-Watson process with immigration,” J. Roy. Statist. Soc., Ser. B,32, No. 1, 149–152 (1970).
R. Bojanic and E. Seneta, “Slowly varying functions and asymptotic relations,” J. Math. Anal. Appl.,34, No. 3, 302–315 (1971).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 727–736, May, 1977.
The author thanks A. M. Zubkov for the formulation of the problem and valuable advice.
Rights and permissions
About this article
Cite this article
Vatutin, V.A. A conditional limit theorem for a critical Branching process with immigration. Mathematical Notes of the Academy of Sciences of the USSR 21, 405–411 (1977). https://doi.org/10.1007/BF01788239
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01788239