Maximal coupling and $V$-stability of discrete nonhomogeneous Markov chains
Authors:
V. V. Golomozyĭ and M. V. Kartashov
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 93 (2016), 19-31
MSC (2010):
Primary 60J45; Secondary 60A05, 60K05
DOI:
https://doi.org/10.1090/tpms/992
Published electronically:
February 7, 2017
MathSciNet review:
3553437
Full-text PDF Free Access
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Additional Information
Abstract: Two time-nonhomogeneous discrete Markov chains whose one-step transition probabilities are close in the $V$-variation norm are considered. The problem of stability of the expectation of $f(X_n)$ with $|f|\le V$ is studied for Markov chains. The main assumption imposed on a Markov chain is the $V$-mixing. The proofs are based on the maximal coupling procedure that maximize the one-step coupling probabilities.
References
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- V. V. Golomozyĭ, M. V. Kartashov, and Yu. M. Kartashov, The impact of stress factor on the price of widow’s pension. Proofs, Teor. Ĭmovir. Mat. Stat. 92 (2015), 23–27; English transl. in Theory Probab. Math. Statist. 92 (2016), 17–22.
- 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
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/TViMS, Utrecht/Kiev, The Netherlands/Ukraine, 1996. MR 1451375
- N. V. Kartashov, Exponential asymptotics of matrices of the Markov renewal, Asymptotic Problems for Stochastic Processes, Preprint 77-24, Akad. Nauk Ukrain. SSR, Inst. Matem., Kiev, 1977, pp. 2-43. (Russian)
- E. Nummelin, A splitting technique for Harris recurrent chains, Z. Wahrscheinlichkeitstheorie and Verw. Geb. 43 (1978), 309–318. MR 0501353
- E. Nummelin and R. L. Tweedie, Geometric ergodicity and R-positivity for general Markov chains, Ann. Probab. 6 (1978), 404–420. MR 0474504
- T. Lindvall, On coupling of discrete renewal sequences, Z. Wahrsch. Verw. Gebiete 48 (1979), 57–70. MR 533006
- I. N. Kovalenko and N. Yu. Kuznetsov, Construction of an embedding renewal process for essentially multidimensional processes of queueing theory, and its application to obtaining limit theorems, Preprint 80-12, Akad. Nauk Ukrain. SSR, Inst. Kibernet., Kiev, 1980. (Russian) MR 612478
- P. Ney, A refinement of the coupling method in renewal theory, Stoch. Process. Appl. 11 (1981), 11–26. MR 608004
- E. Numemelin and P. Tuominen, Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory, Stoch. Proc. Appl. 12 (1982), 187–202. MR 651903
- E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, Cambridge University Press, Cambridge, 1984. MR 776608
- T. Lindvall, Lectures on the Coupling Method, John Wiley and Sons, 1991. MR 1180522
- S. P. Mayn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, 1993. MR 1287609
- P. Tuominen and R. Tweedie, Subgeometric rates of convergence of f-ergodic Markov chains, Adv. Appl. Probab. 26 (1994), 775–798. MR 1285459
- P. Tuominen and R. L. Tweedie, Subgeometric rates of convergence of f-ergodic Markov chains, Adv. Appl. Probab. 26 (1994), 775–798. MR 1285459
- H. Thorisson, Coupling, Stationarity, and Regeneration, Springer, New York, 2000. MR 1741181
- S. F. Jarner and G. O. Roberts, Polynomial convergence rates of Markov chains, Ann. Appl. Probab. 12 (2001), 224–247. MR 1890063
- 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
- R. Douc, E. Mouliness, and J. S. Rosenthal, Quantitative bounds on convergence of Time-nonhomogeneous Markov chains, Ann. Appl. Probab. 14 (2004), no. 4, 1643–1665. MR 2099647
- 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
- R. Douc, E. Moulines, and P. Soulier, Computable convergence rates for subgeometrically ergodic Markov chains, Bernoulli 13 (2007), no. 3, 831–848. MR 2348753
- R. Douc, G. Fort, and A. Guillin, Subgeometric rates of convergence of f-ergodic strong Markov processes, Stoch. Process. Appl. 119 (2009), no. 3, 897–923. MR 2499863
- V. V. Golomozyĭ, Stability of time in-homogeneous Markov chains, Visn. Kyiv. Univer. Ser. Phys. Math. 4 (2009), 10–15. (Ukrainian)
- V. V. Golomozyĭ, A subgeometric estimate for the stability of time-homogeneous Markov chains, Teor. Ĭmovir. Mat. Stat. 81 (2010), 31–46; English transl. in Theory Probab. Math. Statist. 81 (2009), 31–45. MR 2667308
- M. V. Kartashov, Boundedness, limits, and stability of solutions of an nonhomogeneous perturbation of a renewal equation on a half-line, Teor. Ĭmovir. Mat. Stat. 81 (2009), 65–75; English transl. in Theory Probab. Math. Statist. 81 (2010), 71–83. MR 2667311
- M. V. Kartashov and V. V. Golomozyĭ, The mean coupling time of independent discrete renewal processes, Teor. Ĭmovir. Mat. Stat. 84 (2011), 78–85; English transl. in Theory Probab. Math. Statist. 84 (2012), 79–86. MR 2857418
- M. V. Kartashov and V. V. Golomozyĭ, Maximal coupling procedure and stability of discrete Markov chains. I, Teor. Ĭmovir. Mat. Stat. 86 (2012), 81–92; English transl. in Theory Probab. Math. Statist. 86 (2013), 93–104. MR 3241447
- M. V. Kartashov and V. V. Golomozyĭ, Maximal coupling procedure and stability of discrete Markov chains. II, Teor. Ĭmovir. Mat. Stat. 87 (2012), 58–70; English transl. in Theory Probab. Math. Statist. 87 (2013), 65–78. MR 3241447
- V. V. Golomozyĭ and M. V. Kartashov, On coupling moment integrability for time-nonhomogeneous Markov chains, Teor. Ĭmovir. Mat. Stat. 89 (2014), 1–11; English transl. in Theory Probab. Math. Statist. 89 (2014), 1–12. MR 3235170
- V. V. Golomozyĭ, Inequalities for the coupling time of two nonhomogeneous Markov chains, Teor. Ĭmovir. Mat. Stat. 90 (2014), 39–51; English transl. in Theory Probab. Math. Statist. 90 (2015), 43–56. MR 3241859
- M. V. Kartashov and V. V. Golomozyĭ, Maximal coupling and stability of discrete nonhomogeneous Markov chains, Teor. Ĭmovir. Mat. Stat. 91 (2014), 16–26; English transl. in Theory Probab. Math. Statist. 91 (2015), 17–27. MR 3364120
- V. V. Golomozyĭ, M. V. Kartashov, and Yu. M. Kartashov, The impact of stress factor on the price of widow’s pension. Proofs, Teor. Ĭmovir. Mat. Stat. 92 (2015), 23–27; English transl. in Theory Probab. Math. Statist. 92 (2016), 17–22.
- 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
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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, 6, Kyiv 03127, Ukraine
Email:
mailtower@gmail.com
M. V. Kartashov
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
mailtower@gmail.com
Keywords:
Coupling theory,
coupling method,
maximal coupling,
discrete Markov chains,
stability of distributions,
test functions
Received by editor(s):
March 20, 2015
Published electronically:
February 7, 2017
Additional Notes:
The paper was prepared following the talk at the International Conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
Article copyright:
© Copyright 2017
American Mathematical Society