Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-29T08:14:48.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-stationary distributions of birth-and-death processes

Published online by Cambridge University Press:  01 July 2016

James A. Cavender
James A. Cavender*
Affiliation:
Montana State University, Bozeman
Get access Rights & Permissions [Opens in a new window]

Abstract

Let qn(t) be the conditioned probability of finding a birth-and-death process in state n at time t, given that absorption into state 0 has not occurred by then. A family {q1(t), q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

Keywords

BIRTH-AND-DEATH PROCESSQUASI-STATIONARY DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 3 , September 1978 , pp. 570 - 586
Copyright
Copyright © Applied Probability Trust 1978 

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

Bellman, R. (1970) Methods of Non-Linear Analysis, Vol. I. Academic Press, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Karlin, S. (1969) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. London Math. Soc. (3)13, 337–358.CrossRefGoogle Scholar
Mandl, P. (1960) On the asymptotic behavior of probabilities within classes of states of a homogeneous Markov process (in Russian), Časopis Pešt. Mat. 85, 448–456.Google Scholar
Reid, W. T. (1972) Riccati Differential Equations. Academic Press, New York.Google Scholar
Reuter, G. E. H. and Ledermann, W. (1953) On the differential equations for the transition probabilities of Markov processes with denumerably many states. Proc. Camb. Phil. Soc. 49, 247–262.Google Scholar
Richardson, C. H. (1954) An Introduction to the Calculus of Finite Differences. Van Nostrand, New York.Google Scholar
Seneta, E. (1966) Quasi-stationary behaviour in the random walk with continuous time. Austral. J. Statist. 8, 92–98.Google Scholar
Seneta, E. (1973) Non-negative Matrices, Halsted (Wiley), New York.Google Scholar
Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.Google Scholar