An estimate of the stability for nonhomogeneous Markov chains under classical minorization condition
Author:
V. V. Golomozyĭ
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 88 (2014), 35-49
MSC (2010):
Primary 60J45; Secondary 60A05, 60K05
DOI:
https://doi.org/10.1090/S0094-9000-2014-00917-5
Published electronically:
July 24, 2014
MathSciNet review:
3112633
Full-text PDF Free Access
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Additional Information
Abstract: The stability of time inhomogeneous Markov chains is considered under the classical minorization condition. The key tool for the proofs is a modified coupling method for a pair of two (possibly time inhomogeneous) Markov chains.
References
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- V. V. Golomoziĭ, A subgeometric estimate for the stability of time-homogeneous Markov chains, Teor. Ĭmovīr. Mat. Stat. 81 (2009), 31–45 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 81 (2010), 35–50. MR 2667308, DOI https://doi.org/10.1090/S0094-9000-2010-00808-8
- M. V. Kartashov, Boundedness, limits, and stability of solutions of an inhomogeneous perturbation of a renewal equation on a half-line, Teor. Ĭmovīr. Mat. Stat. 81 (2009), 65–75 (Ukrainian, with English and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 81 (2010), 71–83. MR 2667311, DOI https://doi.org/10.1090/S0094-9000-2010-00811-8
- M. V. Kartashov and V. V. Golomoziĭ, The mean coupling time of independent discrete renewal processes, Teor. Ĭmovīr. Mat. Stat. 84 (2011), 77–83 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 84 (2012), 79–86. MR 2857418, DOI https://doi.org/10.1090/S0094-9000-2012-00855-7
- M. V. Kartashov and V. V. Golomoziĭ, Maximal coupling and stability of discrete Markov chains. I, Teor. Ĭmovīr. Mat. Stat. 86 (2011), 81–91 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 86 (2013), 93–104. MR 2986452, DOI https://doi.org/10.1090/S0094-9000-2013-00891-6
- M. V. Kartashov and V. V. Golomozyĭ, Maximal coupling and stability of discrete Markov chains. II, Teor. Imovir. Mat. Stat. 87 (2012), 47–59. (Ukrainian)
References
- W. Feller, An Introduction to Probability Theory and its Applications, vol. 1, John Wiley & Sons, New York, 1966. MR 0210154 (35:1048)
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht/Kiev, The Netherlands/Ukraine, 1996. MR 1451375 (99e:60150)
- N. V. Kartashov, Exponential asymptotics of matrices of the Markov renewal, Asymptotic Problems for Stochastic Processes, Preprint 77-24, Institute of Mathematics of Academy of Science of Ukraine, Kiev, 1977. (Russian)
- I. N. Kovalenko and N. Ju. Kuznecov, Construction of an embedding removal process for essentially multidimensional processes of queueing theory, and its application to obtaining limit theorems, Preprint 80 [Preprint 80], vol. 12, Akad. Nauk Ukrain. SSR Inst. Kibernet., Kiev, 1980. MR 612478 (82i:60142)
- P. Ney, A refinement of the coupling method in renewal theory, Stochastic Process. Appl. 11 (1981), 11–26. MR 608004 (82d:60169)
- E. Numemelin and P. Tuominen, Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory, Stochastic Process. Appl. 12 (1982), 187–202. MR 651903 (83f:60089)
- E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge Tracts in Mathematics, 83, Cambridge University Press, Cambridge, 1984. MR 776608 (87a:60074)
- V. M. Zolotarev, Modern Theory of Summation of Independent Random Variables, “Nauka”, Moscow, 1986; English transl., VSP, Utrecht, the Netherlands–Tokyo, Japan, 1997.
- S. T. Rachev, The Monge–Kantorovich problem on mass transfer and its applications in stochastics, Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 625–653; English transl. in Theory Probab. Appl. 29 (1984), no. 4, 647–676. MR 773434 (86m:60026)
- T. Lindvall, Lectures on the Coupling Method, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley and Sons, New York, 1992. MR 1180522 (94c:60002)
- S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Communications and Control Engineering Series, Springer-Verlag, 1993. MR 1287609 (95j:60103)
- P. Tuominen and R. Tweedie, Subgeometric rates of convergence of f-ergodic Markov chains, Adv. Appl. Probab. 26 (1994), 775–798. MR 1285459 (95m:60097)
- R. L. Tweedie and J. N. Corcoran, Perfect sampling of ergodic Harris chains, Ann. Appl. Probab. 11 (2001), no. 2, 438–451. MR 1843053 (2002g:60111)
- H. Thorisson, Coupling, Stationarity, and Regeneration, Probability and its Applications (New York), Springer, New York, 2000. MR 1741181 (2001b:60003)
- S. F. Jarner and G. O. Roberts, Polynomial convergence rates of Markov chains, Ann. Appl. Probab. 12 (2001), 224–247. MR 1890063 (2003c:60117)
- R. Douc, E. Moulines, and J. S. Rosenthal, Quantitative bounds for geometric convergence rates of Markov chains, Ann. Appl. Probab. 14 (2004), 1643–1664. MR 2099647 (2005i:60146)
- R. Douc, E. Mouliness, and J. S. Rothenthal, Quantitative bounds on convergence of time-inhomogeneous Markov chains, Ann. Appl. Probab. 14 (2004), no. 4, 1643–1665. MR 2099647 (2005i:60146)
- R. Douc, E. Moulines, and P. Soulier, Practical drift conditions for subgeometric rates of convergence, Ann. Appl. Probab. 14 (2004), no. 4, 1353–1377. MR 2071426 (2005e:60156)
- R. Douc, E. Moulines, and P. Soulier, Computable convergence rates for subgeometrically ergodic Markov chains, Bernoulli 13 (2007), no. 3, 831–848. MR 2348753 (2008j:60172)
- R. Douc, G. Fort, and A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Process. Appl. 119 (2009), no. 3, 897–923. MR 2499863 (2010j:60184)
- V. V. Golomozyĭ, Stability of non-homogeneous Markov chains, Visnyk Kyiv Univ., Ser. Fiz. Mat. Nauk 4 (2009), 10–15. (Ukrainian)
- V. V. Golomozyĭ, A subgeometric estimate for the stability of time-homogeneous Markov chains, Teor. Imovir. Mat. Stat. 81 (2010), 31–46; English transl. in Theory Probab. Math. Statist. 81 (2010), 35–50. MR 2667308 (2011c:60232)
- M. V. Kartashov, Boundedness, limits, and stability of solutions of an inhomogeneous perturbation of a renewal equation on a half-line, Teor. Imovir. Mat. Stat. 81 (2009), 65–75; English transl. in Theory Probab. Math. Statist. 81 (2010), 71–83. MR 2667311 (2011f:60154)
- M. V. Kartashov and V. V. Golomozyĭ, The mean coupling time of independent discrete renewal processes, Teor. Imovir. Mat. Stat. 84 (2011), 78–85; English transl. in Theory Probab. Math. Statist. 84 (2012), 79–86. MR 2857418 (2012f:60306)
- M. V. Kartashov and V. V. Golomozyĭ, Maximal coupling procedure and stability of discrete Markov chains. I, Teor. Imovir. Mat. Stat. 86 (2012), 81–92; English transl. in Theory Probab. Math. Statist. 86 (2013), 93–104. MR 2986452
- M. V. Kartashov and V. V. Golomozyĭ, Maximal coupling and stability of discrete Markov chains. II, Teor. Imovir. Mat. Stat. 87 (2012), 47–59. (Ukrainian)
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Additional Information
V. V. Golomozyĭ
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email:
mailtower@gmail.com
Keywords:
Coupling theory,
coupling method,
maximal coupling,
discrete Markov chains,
stability of distributions
Received by editor(s):
December 26, 2011
Published electronically:
July 24, 2014
Article copyright:
© Copyright 2014
American Mathematical Society