The aim of this paper is to investigate the stability of Markov chains with general state space. We present new conditions for the strong stability of Markov chains after a small perturbation of their transition kernels. Also, we obtain perturbation bounds with respect to different quantities.
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D. Aïssani and N.V. Kartashov, “Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels,” Dokl. Akad. Nauk. Ukr. SSR, 11, 3–5 (1983).
D. Aïssani and N.V. Kartashov, “Strong stability of the embedded Markov chain in an M/G/1 system,” Prob. Theor. Math. Stat., 29, 1–5 (1984).
A. A. Borovkov, Probability Processes in Queuing Theory, Fizmatlit, Moskow (1972).
S. L. Campbell and C.D. Meyer, Generalized Inverses of Linear Transformations, Dover, New York (1991).
G. E. Cho and C.D. Meyer, “Comparison of perturbation bounds for the stationary distribution of a Markov chain,” Linear Algebra Appl., 335, 137–150 (2001).
M. P. Drazin, “Pseudoinverse in associative rings and semigroups,” Am. Math. Mont., 65, 506–514 (1958).
C. W. Groetsch, Generalized Inverses of Linear Operators, Marcel Dekker, New York (1977).
V. V. Kalashnikov, “Analysis of ergodicity of queueing systems by means of the direct method of Lyapunov,” Avtomat. Telemekh., 4, 46–54 (1971).
N. V. Kartashov, “Strong stability of Markov chains,” J. Sov. Math., 34, 1493–1498 (1986).
N. V. Kartashov, “Criteria for ergodicity and stability for Markov chains with common phase space,” Theor. Probab. Math. Stat., 30, 71–89 (1985).
N. V. Kartashov, “Inequalities in theorems of ergodicity and stability for Markov chains with common phase space I,” Theor. Probab. Appl., 30, No. 2, 23–35 (1985).
N. V. Kartashov, Strong Stable Markov Chains. VSP, Utrecht (1996).
T. Kato, Perturbation Theory for Linear Operators, Springer–Verlag, Berlin (1976).
J. G. Kemeny and J. L. Snell, Finite Markov Chains, Van Nostrand, New York (1966).
J.G. Kemeny, J. L. Snell, and A.W. Knapp, Denumerable Markov Chains, Van Nostrand, New York (1966).
S.P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer–Verlag, London (1993).
C.D. Meyer, “The condition of a finite Markov chain and perturbation bounds for the limiting probabilities,” SIAM J. Algebraic Discrete Meth., 1, No. 3, 273–283 (1980).
B. Rabta and D. Aïssani, “Strong stability in an (R, s, S) inventory model,” Int. J. Prod. Econ., 97, No. 2, 159–171 (2005).
B. Rabta and D. Aïssani, “Strong stability and perturbation bounds for Markov chains with discrete state space,” Lin. Algebra Appl., 428, No. 8–9, 1921–1927 (2008).
S.T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley, New York (1989).
H. J. Rossberg, “Über die Verteilung von Wartezeiten,” Math. Nachr., 1/2, No. 30, 1–16 (1965).
P. Schweitzer, “Perturbation theory and finite Markov chains,” J. Appl. Probab., 5, 401–413 (1968).
E. Seneta, “Perturbation of the stationary distribution measured by ergodicity coefficients,” Adv. Appl. Probab., 20, No. 1, 228–230 (1988).
E. Seneta, “Sensitivity analysis, ergodicity coefficients, and rank-one updates for finite Markov chains,” in: Numerical Solution of Markov Chains, W. J. Stewart (ed.), Marcel Dekker, New York (1991), pp. 121–129.
D. Stoyan, “Ein Steitigkeitssatz für Einlinge Watermodelle der Bedienungstheorie,” Math. Oper. Forsch. Stat., 3, No. 2, 103–111 (1977).
D. Stoyan, Comparison Methods for Queues and Other Stochastic Models, Daley, New York (1983).
V. M. Zolotarev, “On the continuity of stochastic sequences generated by recurrent processes,” Theor. Probab. Appl., 20, No. 4, 819–832 (1975).
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Proceedings of the XVIII International Seminar on Stability Problems for Stochastic Models, Zakopane, Poland, May 31–June 5, 2009
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Rabta, B., Aïssani, D. Perturbation Bounds for Markov Chains with General State Space. J Math Sci 228, 510–521 (2018). https://doi.org/10.1007/s10958-017-3640-9
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DOI: https://doi.org/10.1007/s10958-017-3640-9