Properties of the stochastic ordering for discrete distributions and their applications to the renewal sequence generated by a nonhomogeneous Markov chain

Golomozyĭ, V.

doi:10.1090/tpms/1046

Properties of the stochastic ordering for discrete distributions and their applications to the renewal sequence generated by a nonhomogeneous Markov chain

Author: V. V. Golomozyĭ
Translated by: N. N. Semenov
Journal: Theor. Probability and Math. Statist. 97 (2018), 33-43
MSC (2010): Primary 60J45; Secondary 60A05, 60K05
DOI: https://doi.org/10.1090/tpms/1046
Published electronically: February 21, 2019
MathSciNet review: 3745997
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Abstract: The generalized stochastic ordering is studied for which the dominating sequence is not necessarily a probability distribution so that its total mass may exceed unity. We study the stochastic ordering for sums as well as random sums of independent as well as dependent random variables. A stochastic ordering is constructed for the renewal sequence generated by a nonhomogeneous Markov chain. The consideration is restricted to the case of discrete random variables.

References

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
Hermann Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications (New York), Springer-Verlag, New York, 2000. MR 1741181
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
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, Estimate for the expectation of excess of a renewal sequence generated by a nonhomogeneous Markov chain under the condition of the existence of a quadratically integrable majorant, Teor. Ĭmovīr. Mat. Stat. 94 (2016), 50–59 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 94 (2017), 53–62. MR 3553453, DOI https://doi.org/10.1090/tpms/1008
Vitaliy Golomoziy, An estimate for an expectation of the simultaneous renewal for time-inhomogeneous Markov chains, Mod. Stoch. Theory Appl. 3 (2016), no. 4, 315–323. MR 3593115, DOI https://doi.org/10.15559/16-VMSTA68
V. V. Golomoziĭ, Estimation of the stability of nonhomogeneous Markov chains under the classical minorization condition, Teor. Ĭmovīr. Mat. Stat. 88 (2013), 32–45 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 88 (2014), 35–49. MR 3112633, DOI https://doi.org/10.1090/S0094-9000-2014-00917-5

V. Golomozyĭ, Stability of time-inhomogeneous Markov chains, Bulletin of Kyiv University (Physics and Mathematical Sciences) 4 (2009), 10–15. (Ukrainian)

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

Achim Klenke and Lutz Mattner, Stochastic ordering of classical discrete distributions, Adv. in Appl. Probab. 42 (2010), no. 2, 392–410. MR 2675109, DOI https://doi.org/10.1239/aap/1275055235