## The Annals of Probability

### The Birth and Death Chain in a Random Environment: Instability and Extinction Theorems

William C. Torrez

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 6, Number 6 (1978), 1026-1043.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995391

Digital Object Identifier
doi:10.1214/aop/1176995391

Mathematical Reviews number (MathSciNet)
MR512418

Zentralblatt MATH identifier
0392.60049

JSTOR