Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T22:20:10.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Subcritical branching processes in a two-state random environment, and a percolation problem on trees

Published online by Cambridge University Press:  14 July 2016

F. M. Dekking
F. M. Dekking*
Affiliation:
Delft University of Technology
*
∗Postal address: Department of Mathematics, Delft University of Technology, Julianalaan 132, 2628 BL Delft, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

We determine the decay rate of the survival probability of subcritical branching processes in a two-state random environment, where one state is subcritical, the other supercritical. This result is applied to obtain the asymptotic behavior (as n →∞) of the number of different words of length n occurring on the binary, and generally the b-ary, tree with Bernoulli percolation.

Keywords

SURVIVAL PROBABILITYBERNOULLI PERCOLATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 4 , December 1987 , pp. 798 - 808
Copyright
Copyright © Applied Probability Trust 1987 

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] Agresti, A. (1975) On the extinction times of varying and random environment branching processes. J. Appl. Prob. 12, 39–46.Google Scholar
[2] Athreya, K. B. and Karlin, S. (1971) Branching processes with random environments, I: extinction probabilities. Ann. Math. Statist. 42, 1499–1520.Google Scholar
[3] Athreya, K. B. and Karlin, S. (1971) Branching processes with random environments II: limit theorems. Ann. Math. Statist. 42, 1843–1858.Google Scholar
[4] Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
[5] Bizley, M. T. L. (1954) Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above the line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line. J. Inst. Actuar. 80, 55–62.CrossRefGoogle Scholar
[6] Smith, W. L. and Wilkinson, W. E. (1969) On branching processes in random environments. Ann. Math. Statist. 40, 814–827.Google Scholar