Another note on the inequality between geometric and p‐generalized arithmetic mean

Christoph Thäle

Mathematische Nachrichten, 2021, vol. 294, issue 10, 1977-1986

Abstract: A central limit theorem and a moderate deviations principle for the ratio of geometric and p‐generalized arithmetic mean are shown. Also a Berry–Esseen‐type upper bound on the rate of convergence in the central limit theorem is proved. This has implications on the probabilistic version of the question of whether the inequality between geometric and p‐generalized arithmetic mean can be reversed or improved up to multiplicative constants. The involved random vectors we study belong to a class of distributions on the ℓpn‐ball introduced by Barthe, Guédon, Mendelson and Naor. The results complement previous limit theorems of Kabluchko, Prochno and Vysotsky.

Date: 2021
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https://doi.org/10.1002/mana.202000238

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