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Improved classical limit analogues for Galton-Watson processes with or without immigration

Published online by Cambridge University Press:  17 April 2009

C.C. Heyde  and
J.R. Leslie
C.C. Heyde
Affiliation:
Department of Statistics, School of General Studies, Australian National University, Canberra, ACT.
J.R. Leslie
Affiliation:
Department of Statistics, School of General Studies, Australian National University, Canberra, ACT.
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Abstract

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It has recently emerged that the central limit theorem and iterated logarithm law for random walk processes have natural counterparts for Galton-Watson processes with or without immigration. Much of the work on these counterparts has previously involved the imposition of supplementary moment conditions. In this paper we show how to dispense with these supplementary conditions and in so doing make the analogy with the random walk results complete.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 2 , October 1971 , pp. 145 - 155
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Bühler, Wolfgang J., “Ein zentraler Grenzwertsatz für Verzweigungsprozesse”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 11 (1969), 139–141.CrossRefGoogle Scholar
[2]Harris, Theodore E., The theory of branching processes (Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 119. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1963; Prentice-Hall, Englewood Cliffs, New Jersey, 1963).CrossRefGoogle Scholar
[3]Heyde, C.C., “On the influence of moments on the rate of convergence to the normal distribution”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 8 (1967), 12–18.CrossRefGoogle Scholar
[4]Heyde, C.C., “Some properties of metrics in a study on convergence to normality”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 11 (1969), 181–192.CrossRefGoogle Scholar
[5]Heyde, C.C., “Some central limit analogues for super-critical Galton-Watson processes”, J. Appl. Probability 8 (1971) 52–59.CrossRefGoogle Scholar
[6]Heyde, C.C., “Some almost sure convergence theorems for branching processes”, Z. Wahrscheinlichkeitstheorie verw. Gebiete (to appear).Google Scholar
[7]Heyde, C.C. and Brown, B.M., “An invariance principle and some convergence rate results for branching processes”, Z. Wahrscheinlichkeitstheorie Verw. Gebiete (to appear).Google Scholar
[8]Heyde, C.C. and Seneta, E., “Analogues of classical limit theorems for the supercritical Galton-Watson process with immigration”, Math. Biosci. (to appear).Google Scholar
[9]Seneta, E., “A note on the supercritical Galton-Watson process with immigration”, Math. Biosci. 6 (1970), 305–312.CrossRefGoogle Scholar