Yet another note on the arithmetic-geometric mean inequality

Abstract

It 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 p -sphere, and also show that sampling the point according to either the cone probability measure on p or the uniform distribution on the ball enclosed by 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.

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

“…is the usual 𝓁 𝑝 -(semi-)norm in ℝ 𝑛 . We argue that the same central limit behaviour as observed in [10] is actually valid for a large class of distributions on 𝔹 𝑛 𝑝 , providing thereby another self-contained proof of the central limit theorem from [10] (Theorem 1.1). Moreover, for this class of distributions we complement the asymptotic results in [10] by adding an upper bound on the speed of convergence (Theorem 1.3) and a moderate deviations principle (Theorem 1.2).…”

“…We argue that the same central limit behaviour as observed in [10] is actually valid for a large class of distributions on 𝔹 𝑛 𝑝 , providing thereby another self-contained proof of the central limit theorem from [10] (Theorem 1.1). Moreover, for this class of distributions we complement the asymptotic results in [10] by adding an upper bound on the speed of convergence (Theorem 1.3) and a moderate deviations principle (Theorem 1.2). The latter describes the probabilistic behaviour of the sequence of random variables  𝑛 on scales between that of the central limit theorem and the large deviations principle.…”

“…as already observed in [10]. This means that with probability tending to 1∕2, as 𝑛 → ∞, inequality (1.1) can be improved by the factor 𝑒 𝛼 𝑝 < 1 if random vectors in 𝔹 𝑛 𝑝 with distribution of the form 𝐏 𝑛,𝑝,𝐖 𝑛 are considered.…”

“…for any 𝑡 ∈ ℝ and any sequence of positive real numbers 𝑏 𝑛 satisfying 𝑏 𝑛 → ∞ and 𝑏 𝑛 ∕ √ 𝑛 → 0, as 𝑛 → ∞. This is in contrast to the large deviations behaviour studied in [10,Theorem 1.4], where for the case that 𝑏 𝑛 = √ 𝑛 the limit on the right-hand side of (1.4) appears to be a rather complicated (but explicit) non-quadratic function  𝑝 ( (𝑡 + 1)𝑒 𝛼 𝑝 ) , where  𝑝 is the function from [ Theorem 1.3. Fix 0 < 𝑝 < ∞.…”

“…Later, this has been refined and extended to the case 𝑝 = 1 by Aldaz [1,2]. Kabluchko, Prochno and Vysotsky [10] showed that for general 𝑝 > 0 the suitably centred and normalized sequence of random variables  𝑛 satisfies, as 𝑛 → ∞, a central limit theorem as well as a large deviations principle (for 𝑝 ≥ 1) under the following distributional assumptions on the random vectors 𝑋 (𝑛) : either 𝑋 (𝑛) is distributed according to the normalized cone or surface measure on the (𝑛 − 1)-dimensional unit 𝓁 𝑝 -sphere 𝕊 𝑛−1 𝑝 or 𝑋 (𝑛) follows a uniform distribution on the 𝑛-dimensional unit 𝓁 𝑝 -ball 𝔹 𝑛 𝑝 . Here, for 0 < 𝑝 < ∞, we write…”