Khinchin-type inequalities via Hadamard's factorisation

TL;DR

This work addresses sharp Khinchin-type inequalities for sums of type $\mathscr{L}'$ random variables by leveraging Hadamard’s factorisation and Newton’s inequalities to prove log-concavity of even-moment sequences. It provides a unified moment-comparison framework that yields sharp constants: for even $p\le q$, $\|X\|_q \le \frac{\|G\|_q}{\|G\|_p}\,\|X\|_p$, extending from Rademacher sums to general type $\mathscr{L}'$ variables and to Hilbert-space-valued sums, with extensions to ferromagnetic dependencies via Lee–Yang theory. The paper also characterizes a broad class of type $\mathscr{L}$ random variables, develops a p$\ge$3 bound under Gaussian-dominated mixtures, and clarifies the landscape of related notions—ultra sub-Gaussianity and strong log-concavity—establishing their equivalence and identifying their boundaries. Together, these results provide sharp, structurally grounded moment inequalities and a detailed comparison of probabilistic regularity notions relevant to Khinchin-type analyses and statistical mechanics models.

Abstract

We prove Khinchin-type inequalities with sharp constants for type L random variables and all even moments. Our main tool is Hadamard's factorisation theorem from complex analysis, combined with Newton's inequalities for elementary symmetric functions. Besides the case of independent summands, we also treat ferromagnetic dependencies in a nonnegative external magnetic field (thanks to Newman's generalisation of the Lee-Yang theorem). Lastly, we compare the notions of type L, ultra sub-Gaussianity (introduced by Nayar and Oleszkiewicz) and strong log-concavity (introduced by Gurvits), with the latter two being equivalent.

Figures (1)

• Figure 1: Implications between ultra sub-Gaussian (USG), strongly log-concave (SLC) and type $\mathscr{L}$ random variables.

Theorems & Definitions (20)

• Theorem 1
• Remark 2
• Corollary 3
• Corollary 4
• Theorem 5
• proof
• Lemma 6
• proof
• Theorem 7
• proof