Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T08:47:25.286Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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