An inequality for the coupling moment in the case of two inhomogeneous Markov chains

Golomozyĭ, V.

doi:10.1090/tpms/948

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 | References | Similar Articles | 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

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

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