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Yet another note on the arithmetic-geometric mean inequality
journal contribution
posted on 2023-06-09, 17:49 authored by Zakhar Kabluchko, Joscha Prochno, Vladislav VysotskiyVladislav VysotskiyIt was shown by E. Gluskin and V.D. Milman in [GAFA Lecture Notes in Math. 1807, 2003] that the classical arithmetic-geometric mean inequality can be reversed (up to a multiplicative constant) with high probability, when applied to coordinates of a point chosen with respect to the surface unit measure on a high-dimensional Euclidean sphere. We present here two asymptotic refinements of this phenomenon in the more general setting of the surface probability measure on a high-dimensional $\ell_p$-sphere, and also show that sampling the point according to either the cone probability measure on $\ell_p$ or the uniform distribution on the ball enclosed by $\ell_p$ yields the same results. First, we prove a central limit theorem, which allows us to identify the precise constants in the reverse inequality. Second, we prove the large deviations counterpart to the central limit theorem, thereby describing the asymptotic behavior beyond the Gaussian scale, and identify the rate function.
History
Publication status
- Published
File Version
- Accepted version
Journal
Studia MathematicaISSN
0039-3223Publisher
Polskiej Akademii Nauk, Instytut MatematycznyExternal DOI
Volume
253Page range
39-55Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2019-05-16First Open Access (FOA) Date
2019-05-17First Compliant Deposit (FCD) Date
2019-05-15Usage metrics
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