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Generalizations of some concentration inequalities

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  • Bhat, M. Ashraf
  • Kosuru, G. Sankara Raju

Abstract

For a real-valued measurable function f and a nonnegative, nondecreasing function ϕ, we first obtain a Chebyshev type inequality which provides an upper bound for ϕ(λ1)μ({x∈Ω:f(x)≥λ1})+∑k=2nϕ(λk)−ϕ(λk−1)μ({x∈Ω:f(x)≥λk}), where 0<λ1<λ2<⋯<λn<∞. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé–Chebyshev, Cantelli and Hoeffding inequalities are obtained.

Suggested Citation

  • Bhat, M. Ashraf & Kosuru, G. Sankara Raju, 2022. "Generalizations of some concentration inequalities," Statistics & Probability Letters, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:stapro:v:182:y:2022:i:c:s0167715221002601
    DOI: 10.1016/j.spl.2021.109298
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    References listed on IDEAS

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    1. Haruhiko Ogasawara, 2019. "The multiple Cantelli inequalities," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(3), pages 495-506, September.
    2. Haruhiko Ogasawara, 2021. "Improvements of the Markov and Chebyshev inequalities using the partial expectation," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(1), pages 116-131, January.
    3. Haruhiko Ogasawara, 2020. "The multivariate Markov and multiple Chebyshev inequalities," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(2), pages 441-453, January.
    4. Eisenberg, Bennett & Ghosh, B. K., 2001. "A generalization of Markov's inequality," Statistics & Probability Letters, Elsevier, vol. 53(1), pages 59-65, May.
    5. Budny, Katarzyna, 2014. "A generalization of Chebyshev’s inequality for Hilbert-space-valued random elements," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 62-65.
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    Cited by:

    1. Roxana A. Ion & Chris A. J. Klaassen & Edwin R. van den Heuvel, 2023. "Sharp inequalities of Bienaymé–Chebyshev and Gauß type for possibly asymmetric intervals around the mean," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 566-601, June.

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