Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T02:30:24.139Z Has data issue: false hasContentIssue false
  • English
  • Français

Some limit theorems for the total progeny of a branching process

Published online by Cambridge University Press:  01 July 2016

A. G. Pakes
A. G. Pakes*
Affiliation:
Monash University
Get access Rights & Permissions [Opens in a new window]

Extract

We consider a branching process in which each individual reproduces independently of all others and has probability aj(j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 < a0, a0 + a1 < 1.

Type
Research Article
Information
Advances in Applied Probability , Volume 3 , Issue 1 , Spring 1971 , pp. 176 - 192
Copyright
Copyright © Applied Probability Trust 1971 

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

Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and their Applications. McGraw-Hill, New York.Google Scholar
Dwass, M. (1969) The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682–685.CrossRefGoogle Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. 2. Wiley, New York.Google Scholar
Feller, W. (1969) An Introduction to Probability Theory and its Applications. Vol. 1 (3rd ed.). Wiley, New York.Google Scholar
Good, I. J. (1949) The number of individuals in a cascade process. Proc. Camb. Phil. Soc. 45, 360–363.CrossRefGoogle Scholar
Harris, T. E. (1948) Branching processes. Ann. Math. Statist. 19, 474–494.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.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
Heyde, C. C. (1970) Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist. 41, 739–742.CrossRefGoogle Scholar
Kendall, D. G. (1948) On the generalized birth and death process. Ann. Math. Statist. 19, 1–15.Google Scholar
Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton-Watson process with mean one and finite variance. Teor. Veroyat. Primen. 11, 579–611.Google Scholar
Lukacs, E. (1960) Characteristic Functions. Griffin, London.Google Scholar
Mullikin, T. W. (1968) Branching processes in neutron transport theory. In Probabilistic Methods in Applied Mathematics. Vol. 1. Bharucha-Reid, A. (Ed.). Academic Press. New York.Google Scholar
Otter, R. (1949) The multiplicative process. Ann. Math. Statist. 20, 206–224.Google Scholar
Pakes, A. (1969) Further results on the critical Galton-Watson process with immigration. J. Aust. Math. Soc. To appear.Google Scholar
Pakes, A. (1971 a) On the critical Galton-Watson process with immigration. J. Aust. Math. Soc. To appear.Google Scholar
Pakes, A. (1971 b) Branching processes with immigration. J. Appl. Prob. 8, 32–42.CrossRefGoogle Scholar
Prabhu, N. U. (1965) Stochastic Processes. Macmillan, New York.Google Scholar
Puri, P. S. (1969) Some limit theorems on branching processes and certain related processes. Sankhya A 31, 57–74.Google Scholar
Roberts, G. E. and Kaufman, H. (1966) Table of Laplace Transforms. Saunders, Philadelphia.Google Scholar
Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–43.CrossRefGoogle Scholar
Seneta, E. (1970) On invariant measures for simple branching processes. (Summary). Bull. Aust. Math. Soc. 2, 359–362.Google Scholar
Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.CrossRefGoogle Scholar