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.
Publié le : 1978-12-14
Classification:
Markov chain,
continuous state space,
birth and death chain,
uniform $\phi$-recurrence,
instability,
invariant measure,
60J05,
60J80,
60J10
@article{1176995391,
author = {Torrez, William C.},
title = {The Birth and Death Chain in a Random Environment: Instability and Extinction Theorems},
journal = {Ann. Probab.},
volume = {6},
number = {6},
year = {1978},
pages = { 1026-1043},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995391}
}
Torrez, William C. The Birth and Death Chain in a Random Environment: Instability and Extinction Theorems. Ann. Probab., Tome 6 (1978) no. 6, pp. 1026-1043. http://gdmltest.u-ga.fr/item/1176995391/