Abstract
The purpose of this article is to present the concept of the character of a finite irreducible Markov chain. It is demonstrated how the sensitivity of the stationary probabilities to perturbations in the transition probabilities can be gauged by the use of the character.
This work was supported in part by the National Science Foundation under grants DMS-9020915 and DDM-8906248.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barlow, J.L. Error bounds and condition estimates for the computation of null vectors with Applications to Markov chains. Preprint, Computer Sci. Dept., Penn State U., pp.1–29 1991.
Berman, A. and Plemmons, R.J., Nonnegative Matrices In The Mathematical Sciences, Academic Press, New York 1979.
Campbell, S.L. and Meyer, C.D., Generalized Inverses Of Linear Transformations, Dover Publications, New York, (1979 edition by Pitman Pub. Ltd., London) 1991.
Chatelin, F. and Miranker, W.L., Acceleration by aggregation of successive approximation methods Linear Algebra and Its Applications, 43(1982),pp.17–47.
Funderlic, R.E., and Meyer, C.D. (1986). Sensitivity of the stationary distribution vector for an ergodic Markov chain, Linear Algebra and Its Applications, 76(1986), pp.1–17.
Golub, G.H. and Meyer, C.D., (1986). Using the QR factorization and group inversion to compute, differentiate, and estimate the sensitivity of stationary probabilities for Markov chains, SIAM J. On Algebraic and Discrete Methods, Vol. 7, No. 2, pp.273–281, 1986.
Grassmann, W.K., Taksar, M.I., and Heyman, D.P., Regenerative analysis and steady state distributions for Markov chains em Operations Research, 33(1985), pp.1107–1116.
Harrod, W.J. and Plemmons, R.J., Comparison of some direct methods for computing stationary distributions of Markov chains, SIAM J. Sci. Statist. Cornput., 5(1984), pp.453–469.
Haviv, M., Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain, SIAM J. Numer. Anal., 22(1987), pp.952–966.
Horn, R.A. and Johnson, C.R., em Topics In Matrix Analysis, Cambridge University Press, Cambridge, 1991.
Kemeny, J.G. and Snell, J.L., Finite Markov Chains, D. Van Nostrand, New York 1960.
Meyer, C.D., and Stewart, G.W., Derivatives and perturbations of eigenvectors, SIAM J. On Numerical Analysis, Vol. 25, No. 3, pp.679–691, 1988.
Meyer, C.D., The role of the group generalized inverse in the theory of finite Markov chains, SIAM Review, Vol. 17, No. 3, pp.443–464, 1975.
Meyer, C.D., The condition of a finite Markov chain and perturbation bounds for the limiting probabilities, SIAM J. On Algebraic and Discrete Methods, Vol. 1, No. 3, pp.273–283, 1980.
Meyer, C.D., Sensitivity of Markov chains, NCSU Technical Report, October, 1991, pp.1–18.
Schweitzer, P.J., Perturbation theory and finite Markov chains, J. Appl. Prob., 5(1968), pp.401–413.
Stewart, G.W. and Zhang, G., On a direct method for the solution of nearly uncoupled Markov chains, Numerische Mathematik, 59(1991), pp.1–11.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Meyer, C.D. (1993). The Character of a Finite Markov Chain. In: Meyer, C.D., Plemmons, R.J. (eds) Linear Algebra, Markov Chains, and Queueing Models. The IMA Volumes in Mathematics and its Applications, vol 48. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8351-2_4
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8351-2_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8353-6
Online ISBN: 978-1-4613-8351-2
eBook Packages: Springer Book Archive