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December, 1978 The Birth and Death Chain in a Random Environment: Instability and Extinction Theorems
William C. Torrez
Ann. Probab. 6(6): 1026-1043 (December, 1978). DOI: 10.1214/aop/1176995391


Let $(Y_n)$ be a recurrent Markov chain with discrete or continuous state space. A model of a birth and death chain $(Z_n)$ controlled by a random environment $(Y_n)$ is formulated wherein the bivariate process $(Y_n, Z_n)$ is taken to be Markovian and the marginal process $(Z_n)$ is a birth and death chain on the nonnegative integers with absorbing state $z = 0$ when a fixed sequence of environmental states of $(Y_n)$ is specified. In this paper, the property of uniform $\phi$-recurrence of $(Y_n)$ is used to prove that with probability one the sequence $(Z_n)$ does not remain positive or bounded. An example is given to show that uniform $\phi$-recurrence of $(Y_n)$ is required to insure this instability property of $(Z_n)$. Conditions are given for the extinction of the process $(Z_n)$ when (i) $(Z_n)$ possesses homogeneous transition probabilities and $(Y_n)$ possesses an invariant measure on discrete state space, and (ii) $(Z_n)$ possesses nonhomogeneous transition probabilities and $(Y_n)$ has general state space.


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William C. Torrez. "The Birth and Death Chain in a Random Environment: Instability and Extinction Theorems." Ann. Probab. 6 (6) 1026 - 1043, December, 1978.


Published: December, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60049
MathSciNet: MR512418
Digital Object Identifier: 10.1214/aop/1176995391

Primary: 60J05
Secondary: 60J10 , 60J80

Keywords: Birth and death chain , continuous state space , instability , invariant measure , Markov chain , uniform $\phi$-recurrence

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 6 • December, 1978
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