New perturbation bounds for denumerable Markov chains

Linear Algebra and Its Applications

Volumn 432, Issue 7, 2010, Pages 1627-1649

  Mouhoubi, Zahir ^a   Aïssani, Djamil ^a  

^a UNIVERSITY OF BEJAIA   (Algeria)

Author keywords

Doeblin chains; Norm ergodicity coefficient; Perturbation analysis of Markov chains; Perturbation bounds; Regular chains; Residual matrix

Indexed keywords

DOEBLIN CHAINS; ERGODICITY; MARKOV CHAIN; NORM ERGODICITY COEFFICIENT; PERTURBATION ANALYSIS; PERTURBATION BOUNDS; RESIDUAL MATRICES;

MARKOV PROCESSES;

EID: 74449093183     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2009.11.020     Document Type: Article

References (43)

1. Ai{dotless}̈ssani D., and Kartashov N.V. Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels. Dokl. Akad. Nauk Ukr. SSR Ser. A 11 (1983) 3-5
2. Altman E., Avrachenkov K.E., and Nunez-Queija R. Perturbation analysis for denumerable Markov chains with applications to queueing models. Adv. Appl. Probab. 36 (2004) 839-853
3. Cho G.E., and Meyer C.D. Markov chain sensitivity measured by mean first passage time. Linear Algebra Appl. 316 (2000) 21-28
4. Cho G.E., and Meyer C.D. Comparison of perturbation bounds for the stationary probabilities of a finite Markov chain. Linear Algebra Appl. 335 (2001) 137-150
5. Funderlic R.E., and Meyer C.D. Sensitivity of the stationary vector for an ergodic Markov chains. Linear Algebra Appl. 76 (1986) 1-17
6. Golub G.H., and Meyer C.D. Using the QR factorization and group inversion to compute, differentiate, and estimate the sensitivity of stationary probabilities for Markov chains. SIAM J. Algebraic Discrete Methods 7 2 (1986) 273-281
7. Hartfiel D.J. How a row change in a stochastic matrix affects the corresponding stochastic eigenvalues?. Linear and Multilinear Algebra 10 (1981) 205-216
8. Haviv M., and van der Heyden L. Perturbation bounds for the stationary probabilities of a finite Markov chains. Adv. in Appl. Probab. 16 (1984) 804-818
9. Heidergott B., and Hordijk A. Taylor series expansions for stationary Markov chains. Adv. in Appl. Probab. 23 (2003) 1046-1070
10. Heidergott B., Hordijk A., and Weisshaupt H. Measure-valued differentaion for stationary Markov chains. Math. Oper. Res. 31 (2006) 154-172
11. Heidergott B., Hordijk A., and van Uitert M. Series expansions for finite-state Markov chains. Probab. Engrg. Inform. Sci. 21 (2007) 381-400
12. B. Heidergott, A. Hordijk, N. Leder, Series expansions for continuous-time Markov processes, Oper. Res., doi:10.1287/opre.1090.0738.
13. Hunter J.J. Mixing times with applications to perturbed Markov chains. Linear Algebra Appl. 417 (2006) 108-123
14. Hunter J.J. Stationary distributions and mean first passage times in Markov chains using generalized inverses. Asia-Pacific J. Oper. Res. 9 (1992) 145-153
15. Hunter J.J. Stationary distributions and mean first passage times of perturbed Markov chains. Linear Algebra Appl. 410 (2005) 217-243
16. Ipsen I.C.F., and Meyer C.D. Uniform stability of Markov Chains. SIAM J. Matrix Anal. Appl. 15 4 (1994) 1061-1074
17. Kartashov N.V. Strong Stable Markov Chains. VSP ed. (1996), Utrecht TBIMC Scientific Publishers, Kiev
18. Kemeny J.G., and Snell J.L. Finite continuous Markov time chains. Theory Probab. Appl. 6 (1961) 101-105
19. Kemeny J.G., and Snell J.L. Finite Markov Chains. The University Series in Undergraduate Mathematics (1960), D. Van Nostrand Co., Inc., Princeton, N.J.-Toranto-London-New York
20. Kemeny J.G., Snell J.L., and Knepp A.W. Denumerable Markov Chains (1976), Springer-Verlag, Berlin
21. Kirkland S.J. Conditioning properties of the stationary distribution for a Markov chain. Electron. J. Linear Algebra 10 (2003) 1-15
22. Kirkland S.J. On question concerning condition numbers for Markov chains. SIAM J. Matrix Anal. Appl. 23 (2002) 1109-1119
23. Kirkland S.J., Neumann M., and Shader B.L. Application of Paz's inequality to perturbations for Markov chains. Linear Algebra Appl. 268 (1998) 183-196
24. Meyer C.D. Analysis of finite Markov chains by group inversion techniques. In: Campbell S.L. (Ed). Recent application of generalized inverses. Res. Notes Math. 68 (1982) 50-81
25. Meyer C.D. Sensitivity of the stationary distribution of the Markov chains. SIAM J. Matrix Anal. Appl. 15 3 (1982) 715-728
26. Meyer C.D. The condition of finite Markov chains and perturbation bounds for the limiting distribution. SIAM J. Algebraic Discrete Methods 1 (1980) 273-287
27. Meyer C.D. The role of the group generalize inverse in the theory of finite Markov chains. SIAM Rev. 17 (1975) 443-464
28. Meyer C.D., and Stewart G.W. Derivatives and perturbations of eigenvectors. SIAM J. Numer. Anal. 25 (1988) 679-691
29. Meyer C.D. The Character of a Finite Markov Chain. Math. Appl. IMA vol. 84 (1993), Springer-Verlag, New York pp. 47-58
30. Meyn S.P., and Tweedie R.L. Markov Chains and Stochastic Stability (1993), Springer-Verlag
31. Meyn S.P., and Tweedie R.L. Computable bounds for geometric convergence rates of Markov Chains. Ann. Appl. Probab. 4 (1994) 981-1011
32. Mouhoubi Z., and Ai{dotless}̈ssani D. Some inequalities of the uniform ergodicity and strong stability of homogeneous Markov chains. Pliska Stud. Math. Bulgar. 17 (2005) 171-189
33. Neuts M.F. Matrix-Geometric Solution in Stochastic Models: An Algorithm Approach (1981), John Hopkins University Press, Baltimore, MD
34. Paz A. Introduction to Probabilistic Automata (1971), Academic, New York
35. Schweitzer P.J. Perturbation theory and finite Markov chains. J. Appl. Probab. 5 (1968) 401-413
36. Seneta E. Application of ergodicity coefficients to homogeneous Markov chains. Contemp. Math. 149 (1993) 189-199
37. Seneta E. Coefficients of ergodicity: structure and applications. Adv. Appl. Probab. 11 (1979) 576-590
38. Seneta E. Perturbation of the stationary distribution measured by the ergodicity coefficients. Adv. Appl. Probab. 20 (1988) 228-230
39. Seneta E. Sensitivity analysis, ergodicity coefficient, and rank-one updates for finite Markov chain. In: Stewart W.J. (Ed). Numerical Solution of Markov Chains. Probability: Pure and Applied (1991), Marcel-Dekker, New York 121-129
40. Seneta E. Sensitivity to perturbation of the stationary distribution: some refinements. Linear Algebra Appl. 108 (1988) 121-126
41. Seneta E. Sensitivity of a finite Markov chains under perturbation. Statist. Probab. Lett. 17 (1993) 163-168
42. Syski R. Ergodic potential. Stochastic Process. Appl. 7 (1978) 311-336
43. Syski R. Perturbation models. Stochastic Process. Appl. 5 (1977) 93-130