An inequality for the coupling moment in the case of two inhomogeneous Markov chains
Author:
V. V. Golomozyĭ
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 90 (2015), 43-56
MSC (2010):
Primary 60J45; Secondary 60A05, 60K05
DOI:
https://doi.org/10.1090/tpms/948
Published electronically:
August 6, 2015
MathSciNet review:
3241859
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: We consider discrete Markov chains with phase space $\{0,1,\dots \}$ and study conditions under which the expectation of the first coupling moment for two independent discrete time inhomogeneous Markov chains is finite. The coupling moment is defined as the first time when both chains simultaneously visit the zero state. Some special cases are considered where a bound for the expectation of the coupling moment is available.
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.
- William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
- N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
- 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, pp. 2–43. (Russian)
- E. Nummelin, A splitting technique for Harris recurrent Markov chains, Z. Wahrsch. Verw. Gebiete 43 (1978), no. 4, 309–318. MR 0501353, DOI https://doi.org/10.1007/BF00534764
- E. Nummelin and R. L. Tweedie, Geometric ergodicity and $R$-positivity for general Markov chains, Ann. Probability 6 (1978), no. 3, 404–420. MR 474504, DOI https://doi.org/10.1214/aop/1176995527
- Torgny Lindvall, On coupling of discrete renewal processes, Z. Wahrsch. Verw. Gebiete 48 (1979), no. 1, 57–70. MR 533006, DOI https://doi.org/10.1007/BF00534882
- I. N. Kovalenko and N. Ju. Kuznecov, Postroenie vlozhennogo protsessa vosstanovleniya dlya sushchestvenno mnogomernykh protsessov teorii massovogo obsluzhivaniya i ego primenenie k polucheniyu predel′nykh teorem, Preprint 80 [Preprint 80], vol. 12, Akad. Nauk Ukrain. SSR, Inst. Kibernet., Kiev, 1980 (Russian). MR 612478
- 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
- Esa Nummelin and Pekka Tuominen, Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory, Stochastic Process. Appl. 12 (1982), no. 2, 187–202. MR 651903, DOI https://doi.org/10.1016/0304-4149%2882%2990041-2
- Esa Nummelin, General irreducible Markov chains and nonnegative operators, Cambridge Tracts in Mathematics, vol. 83, Cambridge University Press, Cambridge, 1984. MR 776608
- Vladimir M. Zolotarev, Modern theory of summation of random variables, Modern Probability and Statistics, VSP, Utrecht, 1997. MR 1640024
- 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 (Russian). MR 773434
- 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
- S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. MR 1287609
- Pekka Tuominen and Richard L. Tweedie, Subgeometric rates of convergence of $f$-ergodic Markov chains, Adv. in Appl. Probab. 26 (1994), no. 3, 775–798. MR 1285459, DOI https://doi.org/10.2307/1427820
- Pekka Tuominen and Richard L. Tweedie, Subgeometric rates of convergence of $f$-ergodic Markov chains, Adv. in Appl. Probab. 26 (1994), no. 3, 775–798. MR 1285459, DOI https://doi.org/10.2307/1427820
- J. N. Corcoran and R. L. Tweedie, Perfect sampling of ergodic Harris chains, Ann. Appl. Probab. 11 (2001), no. 2, 438–451. MR 1843053, DOI https://doi.org/10.1214/aoap/1015345299
- Hermann Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications (New York), Springer-Verlag, New York, 2000. MR 1741181
- Søren F. Jarner and Gareth O. Roberts, Polynomial convergence rates of Markov chains, Ann. Appl. Probab. 12 (2002), no. 1, 224–247. MR 1890063, DOI https://doi.org/10.1214/aoap/1015961162
- R. Douc, E. Moulines, and Jeffrey S. Rosenthal, Quantitative bounds on convergence of time-inhomogeneous Markov chains, Ann. Appl. Probab. 14 (2004), no. 4, 1643–1665. MR 2099647, DOI https://doi.org/10.1214/105051604000000620
- R. Douc, E. Moulines, and Jeffrey S. Rosenthal, Quantitative bounds on convergence of time-inhomogeneous Markov chains, Ann. Appl. Probab. 14 (2004), no. 4, 1643–1665. MR 2099647, DOI https://doi.org/10.1214/105051604000000620
- Randal Douc, Gersende Fort, Eric Moulines, and Philippe Soulier, Practical drift conditions for subgeometric rates of convergence, Ann. Appl. Probab. 14 (2004), no. 3, 1353–1377. MR 2071426, DOI https://doi.org/10.1214/105051604000000323
- 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
- D. J. Daley, Tight bounds for the renewal function of a random walk, Ann. Probab. 8 (1980), no. 3, 615–621. MR 573298
- Randal Douc, Gersende Fort, and Arnaud Guillin, Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stochastic Process. Appl. 119 (2009), no. 3, 897–923. MR 2499863, DOI https://doi.org/10.1016/j.spa.2008.03.007
- 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. Golomozyĭ, Stability of non-homogeneous Markov chains, Visnyk Kyiv Univ., Ser. Fiz. Mat. Nauk 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, 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 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
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.
- W. Feller, An Introduction to Probability Theory and its Applications, vol. 1, John Wiley & Sons, New York, 1966. MR 0228020 (37 \#3604)
- N. V. Kartashov, Strong Stable Markov Chains, VSP, Utrecht/TViMS Scientific Publishers, Kiev, 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, pp. 2–43. (Russian)
- E. Nummelin, A splitting technique for Harris recurrent chains, Z. Wahrscheinlichkeitstheorie Verw. Geb. 43 (1978), 309–318. MR 0501353 (58:18732)
- E. Nummelin and R. L. Tweedie, Geometric ergodicity and $R$-positivity for general Markov chains, Ann. Probab. 6 (1978), 404–420. MR 0474504 (57:14143)
- T. Lindvall, On coupling of discrete renewal sequences, Z. Wahrscheinlichkeitstheorie Verw. Geb. 48 (1979), 57–70. MR 533006 (80g:60091)
- I. N. Kovalenko and N. Ju. Kuznecov, Postroenie vlozhennogo protsessa vosstanovleniya dlya sushchestvenno mnogomernykh protsessov teorii massovogo obsluzhivaniya i ego primenenie k polucheniyu predelnykh teorem, Preprint 80, vol. 12, no. 80-12, Akad. Nauk Ukrain. SSR, Inst. Kibernet., Kiev, 1980. (Russian) MR 612478 (82i:60142)
- P. Ney, A refinement of the coupling method in renewal theory, Stoch. 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, Stoch. Process. Appl. 12 (1982), 187–202. MR 651903 (83f:60089)
- E. Nummelin, General Irreducible Markov Chains and Nonnegative Operators, 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. MR 1640024 (99m:60002)
- 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. (Russian) MR 773434 (86m:60026)
- T. Lindvall, Lectures on the Coupling Method, John Wiley and Sons, 1991. MR 1180522 (94c:60002)
- S. P. Mayn and R. L. Tweedie, Markov Chains and Stochastic Stability, 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)
- P. Tuominen and R. L. 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, 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. Moulines, and J. S. Rosenthal, 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)
- D. J. Daley, Tight bounds for the renewal function of a random walk, Ann. Probab. 8 (1980), no. 3, 615–621. MR 573298 (81e:60094)
- 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 (2010j:60184)
- V. V. Golomozyĭ and M. V. Kartashov, On integrability of the coupling moment for time-inhomogeneous Markov chains, Teor. Imovir. Matem. Statist. 89 (2014), 1–12; English transl. in Theor. Probability and Math. Statist. 89 (2014). MR 3235170
- 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 of the stability for time-homogeneous Markov chains, Teor. Imovir. Matem. Statist. 81 (2010), 31–46; English transl. in Theor. Probability and Math. Statist. 81 (2010), 35–50. MR 2667308 (2011c:60232)
- M. V. Kartashov, Boundedness, limits, and stability of solutions of a perturbation of a nonhomogeneous renewal equation on a semiaxis, Teor. Imovir. Matem. Statist. 81 (2009), 65–75; English transl. in Theor. Probability and Math. Statist. 81 (2010), 71–83. MR 2667311 (2011f:60154)
- M. V. Kartashov and V. V. Golomozyĭ, The mean coupling time for independent discrete renewal processes, Teor. Imovir. Matem. Statist. 84 (2011), 78–85; English transl. in Theor. Probability and 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. Matem. Statist. 86 (2012), 81–92; English transl. in Theor. Probability and Math. Statist. 86 (2013), 93–104. MR 2986452
- M. V. Kartashov and V. V. Golomozyĭ, Maximal coupling procedure and stability of discrete Markov chains. II, Teor. Imovir. Matem. Statist. 87 (2012), 58–70; English transl. in Theor. Probability and Math. Statist. 87 (2013), 65–78. MR 3241447
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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, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
mailtower@gmail.com
Keywords:
Coupling theory,
coupling method,
maximal coupling,
discrete Markov chains,
stability of distributions
Received by editor(s):
September 1, 2013
Published electronically:
August 6, 2015
Article copyright:
© Copyright 2015
American Mathematical Society