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Regularly varying functions in the theory of simple branching processes

Published online by Cambridge University Press:  01 July 2016

E. Seneta
E. Seneta*
Affiliation:
Australian National University, Canberra
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Abstract

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

Keywords

BIENAYMÉ-GALTON-WATSON PROCESSIMMIGRATIONLIMIT LAWSNORMING CONSTANTSREGULARLY VARYING FUNCTIONSINFINITE MEANINVARIANT MEASURES
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 3 , September 1974 , pp. 408 - 420
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Athreya, K. B. (1971) A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Prob. 8, 589–598.Google Scholar
[2] Athreya, K. B. and Ney, P. E. (1973) Branching Processes. Springer, Berlin.Google Scholar
[3] Bruijn, N. G. de (1959) Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform. Nieuw Arch. Wisk. 7, 20–26.Google Scholar
[4] Coifman, R. (1965) Sur l'unicité des solutions de l'equation d'Abel-Schröder et l'itération continue. J. Austral. Math. Soc. 5, 36–47.Google Scholar
[5] Coifman, R. R. and Kuczma, M. (1969) On asymptotically regular solutions of a linear functional equation. Aeq. Math. 2, 332–336.Google Scholar
[6] Darling, D. A. (1970) The Galton-Watson process with infinite mean. J. Appl. Prob. 7, 455–456.Google Scholar
[7] Dubuc, S. (1971) Problèmes relatifs à l'itération de fonctions suggérés par les processus en cascade. Ann. Inst. Fourier (Grenoble) 21, 171–251.Google Scholar
[8] Feller, W. (1966) Introduction to Probability Theory and its Applications. Vol. 2. Wiley, New York.Google Scholar
[9] Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Teor. Veroyat. Primen. 12, 341–346.Google Scholar
[10] Heyde, C. C. (1970) Extension of a result of Seneta for the supercritical Galton-Watson process. Ann. Math. Statist. 41, 739–742.Google Scholar
[11] Kingman, J. F. C. (1965) Stationary measures for branching processes. Proc. Amer. Math. Soc. 16, 245–247.Google Scholar
[12] Krasnosel'skii, M. A. and Rutickii, Ya. B. (1961) Convex Functions and Orlicz Spaces. (transl. by Boron, L. F.) Noordhoff, Groningen.Google Scholar
[13] Kuczma, M. (1964) Note on Schröder's functional equation. J. Austral. Math. Soc. 4, 149–151.CrossRefGoogle Scholar
[14] Kucma, M., (ed.) (1972) Równania Funkcyjne w Teorii Procesów Stochastycznych. Wydawnictwo Uniwersytetu Ślaskiego, 47, Katowice.Google Scholar
[15] Pakes, A. G. (1971) A branching process with a state dependent immigration component Adv. Appl. Prob. 3, 301–314.Google Scholar
[16] Rozmus-Chmura, M. (1968) Les solutions convexes de l'équation fonctionelle g[a (x)] — g(x) = ⊘(x). Publ. Math. 15, 45–48.Google Scholar
[17] Rubin, H. and Vere-Jones, D. (1968) Domains of attraction for the subcritical Galton-Watson process. J. Appl. Prob. 5, 216–219.Google Scholar
[18] Seneta, E. (1968) On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist. 39, 2098–2102.CrossRefGoogle Scholar
[19] Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.Google Scholar
[20] Seneta, E. (1970) On invariant measures for simple branching processes. (Summary.) Bull. Austral. Math. Soc. 2, 359–362.Google Scholar
[21] Seneta, E. (1970) On the supercritical Galton-Watson process with immigration. Math. Biosciences 7, 9–14.Google Scholar
[22] Seneta, E. (1971) On invariant measures for simple branching processes. J. Appl. Prob. 8, 43–51.CrossRefGoogle Scholar
[23] Seneta, E. (1973) The simple branching process with infinite mean, I. J. Appl. Prob. 10, 206–212.Google Scholar
[24] Seneta, E. (1973) A Tauberian theorem of E. Landau and W. Feller. Ann. Prob. 1, 1057–1058.Google Scholar
[25] Slack, R. S. (1968) A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139–145.Google Scholar
[26] Slack, R. S. (1972) Further notes on branching processes with mean 1. Z. Wahrscheinlichkeitsth. 25, 31–38.Google Scholar
[27] Yang, Y. S. (1971) Invariant measures on some Markov processes. Ann. Math. Statist, 42, 1686–1696.Google Scholar