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Hölder's inequality and its reverse—A probabilistic point of view

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  • Lorenz Frühwirth
  • Joscha Prochno

Abstract

In this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in the classical Hölder inequality for random vectors in Rn$\mathbb {R}^n$. We prove a central limit theorem for this ratio, which then allows us to reverse the inequality up to a multiplicative constant with high probability. The models of randomness include the uniform distribution on ℓpn$\ell _p^n$ balls and spheres. We also provide a Berry–Esseen–type result and prove a large and a moderate deviation principle for the suitably normalized Hölder ratio.

Suggested Citation

  • Lorenz Frühwirth & Joscha Prochno, 2023. "Hölder's inequality and its reverse—A probabilistic point of view," Mathematische Nachrichten, Wiley Blackwell, vol. 296(12), pages 5493-5512, December.
    • Handle: RePEc:bla:mathna:v:296:y:2023:i:12:p:5493-5512
    DOI: 10.1002/mana.202200411
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    References listed on IDEAS

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    1. J. M. Aldaz, 2010. "Concentration of the Ratio between the Geometric and Arithmetic Means," Journal of Theoretical Probability, Springer, vol. 23(2), pages 498-508, June.
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