Abstract
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.
Received 23 February, 1971.
Chapter PDF
References
Wolfgang J. Bühler, “Ein zentraler Grenzwertsatz für Verzweigungsprozesse”, Z. Wahrscheinlichkeitstheorie veru. Gebiete 11 (1969), 139–141.
Theodore E. Harris, 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).
C.C. Heyde, “On the influence of moments on the rate of convergence to the normal distribution”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 8 (1967), 12–18.
C.C. Heyde, “Some properties of metrics in a study on convergence to normality”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 11 (1969), 181–192.
C.C. Heyde, “Some central limit analogues mr super-oritical Galton-Watson processes”, J. Appl..probability 8 (1971), 52–59.
C.C. Heyde, “Some almost sure convergence theorems for branching processes”, Z. Wahrscheinlichkeitstheorie verw. Gebiete (to appear).
C.C. Heyde and B.M. Brown, “An invariance principle and some convergence rate results for branching processes”, Z. Wahrscheinlichkeitstheorie verw. Gebiete (to appear).
C.C. Heyde and E. Seneta, “Analogues of classical limit theorems for the supercritical Galton-Watson process with immigration”, Math. Biosei. (to appear).
E. Seneta, “A note on the supercritical Galton-Watson process with immigration”, Math. Biosci. 6 (1970), 305–312.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer New York
About this chapter
Cite this chapter
Heyde, C.C., Leslie, J.R. (2010). Improved classical limit analogues for Galton-Watson processes with or without immigration. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_26
Download citation
DOI: https://doi.org/10.1007/978-1-4419-5823-5_26
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-5822-8
Online ISBN: 978-1-4419-5823-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)