Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T16:28:22.254Z Has data issue: false hasContentIssue false
  • English
  • Français

Calculating extinction probabilities for the birth and death chain in a random environment

Published online by Cambridge University Press:  14 July 2016

William C. Torrez
William C. Torrez*
Affiliation:
New Mexico State University
Get access Rights & Permissions [Opens in a new window]

Abstract

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn) evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space 𝒴 The process (Yn, Zn) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn)-conditioned process (Zn) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

Keywords

BIRTH AND DEATH CHAINEXTINCTION PROBABILITYRANDOM ENVIRONMENTBIVARIATE FINITE DIFFERENCE EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 709 - 720
Copyright
Copyright © Applied Probability Trust 1979 

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

Athreya, K. B. and Karlin, S. (1971a) On branching processes with random environments, I: extinction probabilities. Ann. Math. Statist. 42, 1499–1520.Google Scholar
Athreya, K. B. and Karlin, S. (1971b) On branching processes with random environments, II: limit theorems. Ann. Math. Statist. 42, 1843–1858.CrossRefGoogle Scholar
Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1979) LINPACK Users' Guide. SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Smith, W. (1968) Necessary conditions for almost sure extinction of a branching process with random environment. Ann. Math. Statist. 39, 2136–2140.CrossRefGoogle Scholar
Smith, W. and Wilkinson, W. (1969) On branching processes with random environments. Ann. Math. Statist. 40, 814–827.Google Scholar
Torrez, W. C. (1978) The birth and death chain in a random environment: instability and extinction theorems. Ann. Prob. 6, 1026–1043.Google Scholar
Wilkinson, W. (1969) On calculating extinction probabilities for branching processes in random environments. J. Appl. Prob. 6, 478–492.Google Scholar