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Integer-valued branching processes with immigration

Published online by Cambridge University Press:  01 July 2016

F. W. Steutel*
Affiliation:
University of Technology, Eindhoven
W. Vervaat*
Affiliation:
Catholic University, Nijmegen
S. J. Wolfe*
Affiliation:
University of Delaware
*
Postal address: Department of Mathematics, University of Technology, 5600 MB Eindhoven, The Netherlands.
∗∗Postal address: Mathematisch Instituut, Katholieke Universiteit, Toernooiveld, 6525 ED Nijmegen. The Netherlands.
∗∗∗Postal address: Sharp Laboratory, University of Delaware, Newark, DE 19711, U.S.A. Supported by NSF grants MCS 78-02566 and MCS 80-26546.

Abstract

The notion of self-decomposability for -valued random variables as introduced by Steutel and van Harn [10] and its generalization by van Harn, Steutel and Vervaat [5], are used to study the limiting behaviour of continuous-time Markov branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of random variables obeying a certain difference equation as studied by Vervaat [12] and their continuous-time counterpart considered by Wolfe [13]. An application in queueing theory is indicated. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [14].

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[2] Biggins, J. D. and Shanbhag, D. N. (1981) Some divisibility problems in branching processes. Math. Proc. Camb. Phil. Soc. 90, 321330.CrossRefGoogle Scholar
[3] Foster, J. H. and Williamson, J. S. (1971) Limit theorems for the Galton-Watson process with time-dependent immigration. Z. Wahrscheinlichkeitsth. 20, 227235.Google Scholar
[4] De Haan, L. (1970) On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts 32, Mathematical Centre, Amsterdam.Google Scholar
[5] Van Harn, K., Steutel, F. W. and Vervaat, W. (1982) Self-decomposable discrete distributions and branching processes. Z. Wahrscheinlichkeitsth. 61, 97118.Google Scholar
[6] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[7] Jagers, P. and Nerman, O. (1983) Limit theorems for sums determined by branching and other exponentially growing processes. Stoch. Proc. Appl. Google Scholar
[8] Jurek, Z. J. and Vervaat, W. (1983) An integral representation for self-decomposable Banach space valued random variables. Z. Wahrscheinlichkeitsth. 62, 247262.CrossRefGoogle Scholar
[9] Lukacs, E. (1969) A characterization of stable processes. J. Appl. Prob. 6, 409418.Google Scholar
[10] Steutel, F. W. and Van Harn, K. (1979) Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893899.Google Scholar
[11] Urbanik, K. (1976) Some examples of decomposition semigroups. Bull. Acad. Polon. Sci. Math. 24, 915918.Google Scholar
[12] Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750783.Google Scholar
[13] Wolfe, S. J. (1982) On a continuous analogue of the stochastic difference equation Xn= ρXn-1 + Bn . Stoch. Proc. Appl. 12, 301312.Google Scholar
[14] Yamazato, M. (1975) Some results on infinitely divisible distributions of class L with applications to branching processes. Sci. Rep. Tokyo Kogyo Daigaku, A13, 133139.Google Scholar