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Asymptotic behaviour of immigration-branching processes with general set of types. I: Critical branching part

Published online by Cambridge University Press:  01 July 2016

H. Hering
H. Hering*
Affiliation:
Universität Karlsruhe
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Abstract

We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.

Keywords

IMMIGRATION-BRANCHING PROCESSARBITRARY SET OF TYPESDISCRETE OR CONTINUOUS PARAMETERINHOMOGENEOUS POISSON IMMIGRATION LAWHOMOGENEOUS BRANCHING TRANSITION FUNCTIONGENERATING FUNCTIONALSMOMENT FUNCTIONALSCRITICAL BRANCHING PARTASYMPTOTIC EXTINCTIONRATE OF EXTINCTIONLIMITING DISTRIBUTION FUNCTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 3 , December 1973 , pp. 391 - 416
Copyright
Copyright © Applied Probability Trust 1973 

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References

Arley, N. (1967) On the general birth-and-death-with-immigration stochastic process. Scand. Akt. Tidskr. 50, 175–182.Google Scholar
Bartlett, M. S. (1955) An Introduction to Stochastic Processes with Special Reference to Methods and Applications. Cambridge University Press, Cambridge.Google Scholar
Chistyakov, V. P. (1970a) Some limit theorems for branching processes with immigration. Theor. Probability. Appl. 15, 241–257.CrossRefGoogle Scholar
Chistyakov, V. P. (1970b) Some limit theorems for branching processes with particles of final type. Theor. Probability Appl. 15, 515–521.CrossRefGoogle Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Durham, S. D. (1971) A problem concerning generalized age-dependent branching processes with immigration. Ann. Math. Statist. 42, 1121–1123.CrossRefGoogle Scholar
Foster, J. A. (1969) Branching processes involving immigration. , University of Wisconsin.Google Scholar
Foster, J. A. and Williamson, J. A. (1971) Limit theorems for the Galton-Watson process with time-dependent immigration. Z. Wahrscheinlichkeitsth. 20, 227–235.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin; Prentice-Hall, Englewood Cliffs, N. J. CrossRefGoogle Scholar
Heathcote, C. R. (1965) A branching process allowing immigration. J. R. Statist. Soc. B 27, 138–143.Google Scholar
Heathcote, C. R. (1966) Corrections and comments on the paper “A branching process allowing immigration”. J. R. Statist. Soc. B 28, 213–217.Google Scholar
Hering, H. (1971) Critical Markov branching processes with general set of types. Trans. Amer. Math. Soc. 160, 185–202.CrossRefGoogle Scholar
Hering, H. (1973) Limit theorem for critical branching diffusion processes with absorbing barriers. Math. Biosci. To appear.Google Scholar
Heyde, C. C. and Leslie, J. R. (1971) Improved classical limit analogues for Galton-Watson processes with or without immigration. Bull. Austral. Math. Soc. 5, 145–155.CrossRefGoogle Scholar
Heyde, C. C. and Seneta, E. (1971) Analogues of classical limit theorems for the supercritical Galton-Watson process with immigration. Math. Biosci. 11, 249–259.CrossRefGoogle Scholar
Hille, E. and Phillips, R. S. (1957) Functional Analysis and Semi-Groups. Rev. ed. Amer. Math. Soc. Colloq. Publ., Vol. 31, Amer. Math., Soc., Providence, R. I. Google Scholar
Jagers, P. (1968) Age-dependent branching processes allowing immigration. Theor. Probability Appl. 13, 225–236.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1958) Linear growth, birth and death processes. J. Math. Mech. 4, 643–662.Google Scholar
Kawazu, K. and Watanabe, S. (1971) Branching processes with immigration and related limit theorems. Theor. Probability Appl. 16, 36–54.CrossRefGoogle Scholar
Kingman, J. F. C. (1963) Continuous-time Markov processes. Proc. London Math. Soc. 13, 593–604.CrossRefGoogle Scholar
Lewis, P. A. W. (1969) Asymptotic properties and equilibrium conditions for branching Poisson processes. J. Appl. Prob. 6, 355–371.CrossRefGoogle Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.CrossRefGoogle Scholar
Mullikin, T. W. (1963) Limiting distributions for critical multitype branching processes with discrete time. Trans. Amer. Math. Soc. 106, 469–494.CrossRefGoogle Scholar
Ogura, Y. (1970) Spectral representation for branching processes with immigration on the real half line. Pull. RIMS, Kyoto Univ. 6, 307–321.CrossRefGoogle Scholar
Pakes, A. G. (1970) An asymptotic result for subcritical branching processes with immigration. Bull. Austral. Math. Soc. 2, 223–228.CrossRefGoogle Scholar
Pakes, A. G. (1971a) Branching processes with immigration. J. Appl. Prob. 8, 32–42.CrossRefGoogle Scholar
Pakes, A. G. (1971b) On the critical Galton-Watson process with immigration. J. Austral. Math. Soc. 12, 476–482.CrossRefGoogle Scholar
Pakes, A. G. (1971c) Some results for the supercritical branching process with immigration. Math. Biosci. 11, 355–363.CrossRefGoogle Scholar
Quine, M. P. and Seneta, E. (1969) A limit theorem for the Galton-Watson process with immigration. Austral. J. Statist. 11, 166–173.CrossRefGoogle Scholar
Quine, M. P. (1970) The multitype Galton-Watson process with immigration. J. Appl. Prob. 7, 411–422.CrossRefGoogle Scholar
Reynolds, J. F. (1970) A theorem on discrete time branching processes allowing immigration. J. Appl. Prob. 7, 446–450.CrossRefGoogle Scholar
Savin, A. A. and Chistyakov, V. P. (1962) Some theorems for branching processes with several types of particles. Theor. Probability Appl. 7, 93–100.CrossRefGoogle Scholar
Seneta, E. (1968) The stationary distribution of a branching process allowing immigration: A remark on the critical case. J. R. Statist. Soc. B 30, 176–179.Google Scholar
Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.CrossRefGoogle Scholar
Seneta, E. (1970a) An explicit-limit theorem for the critical Galton-Watson process with immigration. J. R. Statist. Soc. B 32, 149–152.Google Scholar
Seneta, E. (1970b) A note on the supercritical Galton-Watson process with immigration. Math. Biosci. 6, 305–315.CrossRefGoogle Scholar
Seneta, E. (1970c) On the supercritical Galton-Watson process with immigration. Math. Biosci. 7, 9–14.CrossRefGoogle Scholar
Sevast'yanov, B. A. (1951) Theory of branching stochastic processes. Uspehi Mat. Nauk VI.6, 47–99. In Russian.Google Scholar
Sevast'yanov, B. A. (1957) Limit theorems for branching stochastic processes of a special form. Theor. Probability Appl. 2, 321–331.CrossRefGoogle Scholar
Singer, B. (1970) Some asymptotic results in a model of population growth. I. A class of birth and death processes. Ann. Math. Statist. 41, 115–132.CrossRefGoogle Scholar
Singer, B. (1971) Some asymptotic results in a model of population growth. II. Positive recurrent chains. Ann. Math. Statist. 42, 1296–1315.CrossRefGoogle Scholar