Monotonicity of functionals associated to product measures via their Fourier transform and applications

2025/04/08 by Malliaris, Andreas
#FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.06172

Abstract

Let μ be a probability measure on ℝ. We give conditions on the Fourier transform of its density for functionals of the form H(a)=∫_ℝⁿh(⟨ a,x⟩)μⁿ(dx) to be Schur monotone. As applications, we put certain known and new results under the same umbrella, given by a condition on the Fourier transform of the density. These results include certain moment comparisons for independent and identically distributed random vectors, when the norm is given by intersection bodies, and the corresponding vector Khinchin inequalities. We also extend the discussion to higher dimensions.

Related

• Efron's monotonicity property for measures on ℝ²
• Absolutely Continuous Representing Measures of Moment Sequences with an Emphasis on the Regularity of the Densities
• A measure with small support and p-summable Fourier transform
• Regularity of Schrödinger's functional equation in the weak topology and moment measures
• On Multivariate Hyperbolically Completely Monotone Densities and Their Laplace Transforms