Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-05T22:29:12.126Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear growth in near-critical population-size-dependent multitype Galton–Watson processes

Published online by Cambridge University Press:  14 July 2016

Fima C. Klebaner
Fima C. Klebaner*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

Keywords

MULTITYPE BRANCHING PROCESSESMULTIDIMENSIONAL MARKOV CHAINSCRITICALITYGROWTH RATESGAMMA LIMIT
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 3 , September 1989 , pp. 431 - 445
Copyright
Copyright © Applied Probability Trust 1989 

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. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
[2] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[3] Höpfner, R. (1985) On some classes of population-size-dependent Galton-Watson processes. J. Appl. Prob. 22, 25–36.CrossRefGoogle Scholar
[4] Joffe, A. and Spitzer, F. (1967) On multitype branching processes with ρ ≦ 1. J. Math. Anal. Appl. 19, 409–430.Google Scholar
[5] Kesten, H. (1976) Recurrence criteria for multidimensional Markov chains and multidimensional linear birth and death processes. Adv. Appl. Prob. 8, 58–87.Google Scholar
[6] Kersting, G. (1986) On recurrence and transience of growth models. J. Appl. Prob. 23, 614–625.Google Scholar
[7] Klebaner, F. C. (1984) On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.Google Scholar
[8] Mullikin, T. W. (1963) Limiting distributions for critical multitype branching processes with discrete time. Trans. Amer. Math. Soc. 106, 469–494.Google Scholar
[9] Seneta, E. (1981) Nonnegative Matrices and Markov Chains. Springer-Verlag, New York.Google Scholar