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Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains

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  • Chen Jia

    (University of Texas at Dallas)

Abstract

The $$L^p$$ L p maximal inequalities for martingales are one of the classical results in probability theory. Here we establish the sharp moderate maximal inequalities for upward skip-free Markov chains, which include the $$L^p$$ L p maximal inequalities as special cases. Furthermore, we apply our theory to two specific examples and obtain their moderate maximal inequalities: the first one is the M/M/1 queue and the second one is an upward skip-free Markov chain with large death jumps. These two examples have the same total birth and death rates. However, the former exhibits a phase transition phenomenon while the latter does not.

Suggested Citation

  • Chen Jia, 2019. "Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1382-1398, September.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0820-6
    DOI: 10.1007/s10959-018-0820-6
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    References listed on IDEAS

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    1. S. E. Graversen & G. Peškir, 1998. "Optimal Stopping and Maximal Inequalities for Linear Diffusions," Journal of Theoretical Probability, Springer, vol. 11(1), pages 259-277, January.
    2. Yan, Litan & Zhu, Bei, 2004. "A ratio inequality for Bessel processes," Statistics & Probability Letters, Elsevier, vol. 66(1), pages 35-44, January.
    3. James Allen Fill, 2009. "On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains," Journal of Theoretical Probability, Springer, vol. 22(3), pages 587-600, September.
    4. Goran Peskir, 2001. "Bounding the Maximal Height of a Diffusion by the Time Elapsed," Journal of Theoretical Probability, Springer, vol. 14(3), pages 845-855, July.
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