Source: Adv. in Appl. Probab.
Volume 37, Number 4
We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear
equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides
a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with
much more general transition structures to be accommodated. One application is to give a new proof of an important
result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove
new results for downwardly skip-free chains, such as the Markov branching process and several of its many
generalizations. This permits us to establish uniqueness criteria for several models, including the general birth,
death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being
neither upwardly skip-free nor downwardly skip-free.
Anderson, W. J. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.
Brockwell, P. J. (1985). The extinction time of a birth, death and catastrophe process and of a related diffusion model. Adv. Appl. Prob. 17, 42--52.
Mathematical Reviews (MathSciNet): MR778592
Brockwell, P. J. (1986). The extinction time of a general birth and death process with catastrophes. J. Appl. Prob. 23, 851--858.
Mathematical Reviews (MathSciNet): MR867182
Brockwell, P. J., Gani, J. and Resnick, S. I. (1982). Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709--731.
Mathematical Reviews (MathSciNet): MR677553
Chen, A. Y. (2002). Uniqueness and extinction properties of generalized Markov branching processes. J. Math. Anal. Appl. 274, 482--494.
Chen, A. Y. and Renshaw, E. (1990). Markov branching processes with instantaneous immigration. Prob. Theory Relat. Fields 87, 204--240.
Chen, A. Y. and Renshaw, E. (1993). Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Prob. Theory Relat. Fields 94, 427--456.
Chen, M. F. (1992). From Markov Chains to Nonequilibrium Particle Systems. World Scientific, Singapore.
Chen, M. F. (1999). Single birth processes. Chinese Ann. Math. Ser. A 20, 77--82.
Chen, M. F. and Zheng, X. G. (1983). Uniqueness criterion for $q$-processes. Sci. Sinica Ser. A 26, 11--24.
Mathematical Reviews (MathSciNet): MR706062
Chen, R. R. (1997). An extended class of time-continuous branching processes. J. Appl. Prob. 34, 14--23.
Feller, W. (1940). On the integro-differential equations of purely discontinuous Markoff processes. Trans. Amer. Math. Soc. 48, 488--515.
Mathematical Reviews (MathSciNet): MR2697
Hart, A. G. and Pollett, P. K. (1996). Direct analytical methods for determining quasistationary distributions for continuous-time Markov chains. In Athens Conf. on Applied Probability and Time Series Analysis, Vol. 1 (Lecture Notes Statist. 114), eds C. C. Heyde et al., Springer, New York, pp. 116--126.
Hart, A. G. and Pollett, P. K. (2000). New methods for determining quasi-stationary distributions for Markov chains. Math. Comput. Modelling 31, 143--150.
Hou, C. T. (1974). The criterion for uniqueness of a $Q$-process. Sci. Sinica 17, 141--159.
Mathematical Reviews (MathSciNet): MR518020
Hou, Z. T. and Guo, Q. F. (1988). Homogeneous Denumerable Markov Processes. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR960690
Pakes, A. G. (1986). The Markov branching-catastrophe process. Stoch. Process. Appl. 23, 1--33.
Mathematical Reviews (MathSciNet): MR866285
Pollett, P. K. (1991). Invariant measures for $Q$-processes when $Q$ is not regular. Adv. Appl. Prob. 23, 277--292.
Pollett, P. K. (2001). Quasi-stationarity in populations that are subject to large-scale mortality or emigration. Environ. Internat. 27, 231--236.
Pollett, P. K. and Taylor, P. G. (1993). On the problem of establishing the existence of stationary distributions for continuous-time Markov chains. Prob. Eng. Inf. Sci. 7, 529--543.
Reuter, G. E. H. (1957). Denumerable Markov processes and the associated contraction semigroups on $l$. Acta Math. 97, 1--46.
Mathematical Reviews (MathSciNet): MR102123
Reuter, G. E. H. (1976). Denumerable Markov processes. IV. On C. T. Hou's uniqueness theorem for<br/> $Q$-semigroups. Z. Wahrscheinlichkeitsth. 33, 309--315.
Mathematical Reviews (MathSciNet): MR458606
Yan, S. J. and Chen, M. F. (1986). Multidimensional $Q$-processes. Chinese Ann. Math. Ser. A 7, 90--110.
Mathematical Reviews (MathSciNet): MR851278
Zhang, J. K. (1984). Generalized birth--death processes. Acta Math. Sinica 46, 241--259 (in Chinese).
Zhang, Y. H. (2001). Strong ergodicity for single-birth processes. J. Appl. Prob. 38, 270--277.