Hostname: page-component-5c6d5d7d68-wp2c8 Total loading time: 0 Render date: 2024-08-11T16:46:59.863Z Has data issue: false hasContentIssue false
  • English
  • Français

Multitype finite mean supercritical age-dependent branching processes

Published online by Cambridge University Press:  14 July 2016

Harry Cohn
Harry Cohn*
Affiliation:
The University of Melbourne
*
∗ Postal address: Department of Statistics, The University of Melbourne, Parkville, VIC 3052, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {Zφ (t)} be a multitype finite-mean supercritical (CMJ) process with general characteristic φ and malthusian parameter α. It is shown that there exist some constants {c(t} such that {c–1(t)Zφ (t)} converges almost surely to a non-degenerate limit Wφ as t →∞. Characterizations of {c(t)} and Wφ are also given.

Keywords

ALMOST SURE CONVERGENCECRUMP-MODE-JAGERS BRANCHING PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 398 - 403
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] Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhauser, Boston.CrossRefGoogle Scholar
[2] Athreya, K. B. (1970) A simple proof of a result of Kesten and Stigum on supercritical multitype Galton–Watson branching processes. Ann. Math. Statist. 41, 195–202.Google Scholar
[3] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
[4] Cohn, H. (1985) A martingale approach to supercritical (CMJ) branching processes. Ann. Prob. 13, 1179–1191.CrossRefGoogle Scholar
[5] Cohn, H. and Hall, P. (1982) On the limit behaviour of weighted sums of random variables. Z. Wahrscheinlichkeitsth. 59, 319–331.CrossRefGoogle Scholar
[6] Crump, K. S. (1970) On systems of renewal equations. J. Math. Anal. Appl. 30, 425–434.CrossRefGoogle Scholar
[7] Doney, R. A. (1976) On single- and multi-type general age-dependent branching processes. J. Appl. Prob. 13, 239–246.Google Scholar
[8] Hering, H. (1978) The non-degenerate limit for supercritical branching diffusions. Duke Math. J. 45, 561–600.CrossRefGoogle Scholar
[9] Hoppe, F. (1976) Supercritical multitype branching processes. Ann. Prob. 4, 561–600.Google Scholar
[10] Kesten, H. and Stigum, B. P. (1966) A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37, 1211–1223.Google Scholar
[11] Mode, C. J. (1971) Multitype Branching Processes. Elsevier, New York.Google Scholar
[12] Nerman, O. (1979) On the convergence of the supercritical general branching process. Thesis. Department of Mathematics, Chalmers University of Technology and University of Goteborg.Google Scholar
[13] Seneta, E. (1968) On recent theorems concerning the supercritical Galton–Watson process. Ann. Math. Statist. 39, 2098–2102.CrossRefGoogle Scholar