Estimates of stability of transition probabilities for non-homogeneous Markov chains in the case of the uniform minorization
Author:
V. V. Golomozyĭ
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 101 (2020), 85-101
MSC (2020):
Primary 60J45; Secondary 60A05, 60K05
DOI:
https://doi.org/10.1090/tpms/1113
Published electronically:
January 5, 2021
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Additional Information
Abstract: Bounds for the stability of transition probabilities are obtained for two time non-homogeneous Markov chains in discrete time and with a general space of states. The bounds are obtained for the following two cases: first, for the case of the minorization in the whole space, and, second, for the case where transition probabilities of the chains are close each to other. Different estimates of stability are obtained in the paper for different types of closeness.
References
- W. Doeblin, Expose de la theorie des chaines simples constantes de Markov a un nomber fini d’estats, Mathematique de l’Union Interbalkanique 2 (1938), 77–105.
- N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
- Peter Ney, A refinement of the coupling method in renewal theory, Stochastic Process. Appl. 11 (1981), no. 1, 11–26. MR 608004, DOI https://doi.org/10.1016/0304-4149%2881%2990018-1
- Torgny Lindvall, Lectures on the coupling method, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR 1180522
- Torgny Lindvall, On coupling of continuous-time renewal processes, J. Appl. Probab. 19 (1982), no. 1, 82–89. MR 644421, DOI https://doi.org/10.1017/s0021900200028308
- Hermann Thorisson, The coupling of regenerative processes, Adv. in Appl. Probab. 15 (1983), no. 3, 531–561. MR 706616, DOI https://doi.org/10.2307/1426618
- Hermann Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications (New York), Springer-Verlag, New York, 2000. MR 1741181
- Jeffrey S. Rosenthal, Quantitative convergence rates of Markov chains: a simple account, Electron. Comm. Probab. 7 (2002), 123–128. MR 1917546, DOI https://doi.org/10.1214/ECP.v7-1054
- Randal Douc, Eric Moulines, and Philippe Soulier, Subgeometric ergodicity of Markov chains, Dependence in probability and statistics, Lect. Notes Stat., vol. 187, Springer, New York, 2006, pp. 55–64. MR 2283249, DOI https://doi.org/10.1007/0-387-36062-X_2
- Randal Douc, Eric Moulines, and Philippe Soulier, Computable convergence rates for sub-geometric ergodic Markov chains, Bernoulli 13 (2007), no. 3, 831–848. MR 2348753, DOI https://doi.org/10.3150/07-BEJ5162
- V. Golomoziy, Stability of non-homogeneous Markov chains, Vysnik Kyiv. Univ. 4 (2009), 10–15. (Ukrainian)
- 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 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 procedure and stability of discrete Markov chains. II, Theory Probab. Math. Statist. 87 (2013), 65–78. Translation of Teor. Ǐmovīr. Mat. Stat. No. 87 (2012), 58–70. MR 3241447, DOI https://doi.org/10.1090/S0094-9000-2014-00905-9
- V. V. Golomoziy and N. V. Kartashov, On coupling moment integrability for time-inhomogeneous Markov chains, Teor. Ĭmovīr. Mat. Stat. 89 (2013), 1–11 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 89 (2014), 1–12. MR 3235170, DOI https://doi.org/10.1090/S0094-9000-2015-00930-3
- V. V. Golomoziĭ and M. V. Kartashov, Maximal coupling and stability of discrete inhomogeneous Markov chains, Teor. Ĭmovīr. Mat. Stat. 91 (2014), 15–25 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 91 (2015), 17–27. MR 3364120, DOI https://doi.org/10.1090/tpms/963
- Yurij Kartashov, Vitalij Golomoziy, and Nikolai Kartashov, The impact of stress factors on the price of widow’s pensions, Modern problems in insurance mathematics, EAA Ser., Springer, Cham, 2014, pp. 223–237. MR 3330687
- V. Kalashnikov, Estimation of duration of transition regime for complex stochastic systems, in Theory of Compound Systems and Methods of Their Simulation (V. V. Kalashnikov ed.), Moscow, VNIISI, 1980, pp. 63–71. (in Russian)
- David Griffeath, A maximal coupling for Markov chains, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 95–106. MR 370771, DOI https://doi.org/10.1007/BF00539434
- Yurij Kartashov, Vitalij Golomoziy, and Nikolai Kartashov, The impact of stress factors on the price of widow’s pensions, Modern problems in insurance mathematics, EAA Ser., Springer, Cham, 2014, pp. 223–237. MR 3330687
- D. S. Sīl′vestrov, Synchronized regenerative processes and explicit estimates for the rate of convergence in ergodic theorems, Dokl. Akad. Nauk Ukrain. SSR Ser. A 11 (1980), 22–25 (Russian, with English summary). MR 609414
- Dimitriĭ S. Sil′vestrov, Explicit estimates in ergodic theorems for regenerative processes, Elektron. Informationsverarb. Kybernet. 16 (1980), no. 8-9, 461–463 (Russian, with English and German summaries). MR 619339
- D. S. Sil′vestrov, Method of a probability space in ergodic theorems for regenerative processes. I, Math. Operationsforsch. Statist. Ser. Optim. 14 (1983), no. 2, 285–299 (Russian, with English and German summaries). MR 701801, DOI https://doi.org/10.1080/02331938308842856
- Dmitrii S. Silvestrov, Coupling for Markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-Markov switchings, Acta Appl. Math. 34 (1994), no. 1-2, 109–124. MR 1273849, DOI https://doi.org/10.1007/BF00994260
- J. W. Pitman, On coupling of Markov chains, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 4, 315–322. MR 415775, DOI https://doi.org/10.1007/BF00532957
References
- W. Doeblin, Expose de la theorie des chaines simples constantes de Markov a un nomber fini d’estats, Mathematique de l’Union Interbalkanique 2 (1938), 77–105.
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TBiMC, Utrecht/Kiev, 1996. MR 1451375
- P. Ney, A refinement of the coupling method in renewal theory, Stoch. Process. Appl. 11 (1981), 11–26. MR 608004
- T. Lindvall, Lectures on the Coupling Method, John Wiley and Sons, New York, 1991. MR 1180522
- T. Lindvall, On coupling for continuous time renewal processes, J. Appl. Probab. 19 (1982), 82–89. MR 644421
- H. Thorisson, The coupling of regenerative processes, Adv. Appl. Probab. 15 (1983), 531–561. MR 706616
- H. Thorisson, Coupling, Stationarity, and Regeneration, Springer, New York, 2000. MR 1741181
- 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 1917546
- R. Douc, E. Mouliness, and P. Solier, Subgeometric ergodicity of Markov chains, In: P. Bertail, P. Soulier, P. Doukhan (eds.) Dependence in Probability and Statistics. Lecture Notes in Statistics, vol. 187. Springer, New York, 2007, pp. 55–64. MR 2283249
- R. Douc, E. Moulines, and P. Soulier, Computable convergence rates for sub-geometric ergodic Markov chains, Bernoulli 13 (2007), no. 3, 831–848. MR 2348753
- V. Golomoziy, Stability of non-homogeneous Markov chains, Vysnik Kyiv. Univ. 4 (2009), 10–15. (Ukrainian)
- V. Golomoziy, A subgeometric estimate of the stability for time-homogeneous Markov chains, Teor. Imovirnost. Matem. Statyst. 81 (2009), 31–45; English transl. in Theor. Probab. Math. Statist. 81 (2010), 35–50. MR 2667308
- N. Kartashov and V. Golomoziy, Maximal coupling procedure and stability of discrete Markov chains. I, Teor. Imovirnost. Matem. Statyst. 86 (2012), 81–92; English transl. in Theor. Probab. Math. Statist. 86 (2013), 93–104. MR 2986452
- N. Kartashov and V. Golomoziy, Maximal coupling procedure and stability of discrete Markov chains. II, Teor. Imovirnost. Matem. Statyst. 87 (2012), 58–70; English transl. in Theor. Probab. Math. Statist. 87 (2013), 65–78. MR 3241447
- V. Golomoziy and N. Kartashov On coupling moment integrability for time-non-homogeneous Markov chains, Teor. Imovirnost. Matem. Statyst. 89 (2013), 1–11; English transl. in Theor. Probab. Math. Statist. 89 (2014), 1–12. MR 3235170
- N. Kartashov and V. Golomoziy, Maximal coupling and stability of discrete non-homogeneous Markov chains, Teor. Imovirnost. Matem. Statyst. 91 (2014), 15–25; English transl. in Theor. Probab. Math. Statist. 91 (2015), 17–27. MR 3364120
- V. Golomoziy, N. Kartashov, and Y. Kartashov, Impact of the stress factor on the price of widow’s pensions. Proofs, Teor. Imovirnost. Matem. Statyst. 92 (2015), 23–27; English transl. in Theor. Probab. Math. Statist. 92 (2016), 17–22. MR 3330687
- V. Kalashnikov, Estimation of duration of transition regime for complex stochastic systems, in Theory of Compound Systems and Methods of Their Simulation (V. V. Kalashnikov ed.), Moscow, VNIISI, 1980, pp. 63–71. (in Russian)
- D. Griffeath, A maximal coupling for Markov chains, Z. Wahrsch. verw. Gebiete 31 (1975), 95–106. MR 370771
- Y. Kartashov, V. Golomoziy, and N. Kartashov, The impact of stress factor on the price of widow’s pension, Modern Problems in Insurance Mathematics (D. Silverstrov and A. Martin-Löf, eds.), E. A. A. Series, Springer, 2014, pp. 223–237. MR 3330687
- D. Silvestrov, Synchronized regenerative processes and upper estimates for rate of convergence in ergodic theorems, Dopovidi. Acad. Sci. Ukraine, Series A 11 (1980), 22–25. (in Russian) MR 609414
- D. Silvestrov, Upper estimators in ergodic theorems for regenerative processes, Elektron. Inform. Kybernetik 16 (1980), no. 8/9, 461–463. MR 619339
- D. Silvestrov, Method of a single probability space in ergodic theorems for regenerative processes, parts 1–3, Math. Operat. Statist., Ser. Optim.; Part 1 14 (1983), no. 2, 285–299; Part 2 16 (1984), no. 4, 216–231; Part 3 16 (1984), no. 4, 232–244. MR 701801
- D. Silvestrov, Coupling for Markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-Markov switchings, Acta Appl. Math. 34 (1994), 109–124. MR 1273849
- J. W. Pitman, On coupling of Markov chains, Z. Wahrsch. verw. Gebiete 35 (1979), 315–322. MR 415775
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Additional Information
V. V. Golomozyĭ
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
vitaliy.golomoziy@gmail.com
Keywords:
Coupling method,
non-homogeneous Markov chains,
stability of transition probabilities,
minorization in the whole space
Received by editor(s):
September 30, 2019
Published electronically:
January 5, 2021
Article copyright:
© Copyright 2020
American Mathematical Society
