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Stability of Khintchine inequalities with optimal constants between the second and the $p$-th moment for $p \ge 3$

Jacek Jakimiuk

TL;DR

The paper strengthens Khintchine-type inequalities between the second and p-th moments (p≥3) by introducing deficit terms that quantify stability relative to extremal coefficient configurations. It develops two stability notions: Gaussian stability, with a deficit proportional to ∑ a_i^4, and diagonal stability, with a deficit proportional to ∑ (a_i^2 − 1/n)^2, leveraging Schur order and refined convexity of the auxiliary function ψ_s. The results are complemented by concentration bounds for Rademacher sums and a detailed analysis of the function ψ_s and its second derivative, enabling explicit constants C_p and optimality statements. The work not only identifies optimality in the Gaussian case (exponent 4 is necessary) but also provides concrete, computable bounds across p, along with open questions on extensions to other distributions and stability notions. Overall, the findings deepen understanding of how the distribution of coefficients controls higher moments of Rademacher sums and offer precise deficit-driven refinements of classical Khintchine inequalities.

Abstract

We give a strengthening of the classical Khintchine inequality between the second and the $p$-th moment for $p \ge 3$ with optimal constant by adding a deficit depending on the vector of coefficients of the Rademacher sum.

Stability of Khintchine inequalities with optimal constants between the second and the $p$-th moment for $p \ge 3$

TL;DR

The paper strengthens Khintchine-type inequalities between the second and p-th moments (p≥3) by introducing deficit terms that quantify stability relative to extremal coefficient configurations. It develops two stability notions: Gaussian stability, with a deficit proportional to ∑ a_i^4, and diagonal stability, with a deficit proportional to ∑ (a_i^2 − 1/n)^2, leveraging Schur order and refined convexity of the auxiliary function ψ_s. The results are complemented by concentration bounds for Rademacher sums and a detailed analysis of the function ψ_s and its second derivative, enabling explicit constants C_p and optimality statements. The work not only identifies optimality in the Gaussian case (exponent 4 is necessary) but also provides concrete, computable bounds across p, along with open questions on extensions to other distributions and stability notions. Overall, the findings deepen understanding of how the distribution of coefficients controls higher moments of Rademacher sums and offer precise deficit-driven refinements of classical Khintchine inequalities.

Abstract

We give a strengthening of the classical Khintchine inequality between the second and the -th moment for with optimal constant by adding a deficit depending on the vector of coefficients of the Rademacher sum.

Paper Structure

This paper contains 12 sections, 15 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Schur order
  3. Main results
  4. Preliminary results
  5. The function $\psi_s$
  6. Concentration bounds on Rademacher sums
  7. Gaussian stability
  8. Diagonal stability
  9. Final remarks and open questions
  10. Optimality of Theorem \ref{['gauss']}
  11. Estimates on constants $C_p$
  12. Open questions

Key Result

Proposition 1.1

Let $\varepsilon_1, \ldots, \varepsilon_n$ be i.i.d. Rademacher random variables, $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers satisfying $\sum_{i=1}^na_i^2 = \sum_{i=1}^nb_i^2 = 1$ and $(a_1^2, \ldots, a_n^2) \prec (b_1^2, \ldots, b_n^2)$, $\Phi : \mathbb{R} \to \mathbb{R}$ be even and such In particular for $p \ge 3$ we get equivalently the function $x \mapsto -\mathbb{E}\left|\sum_{i=1

Theorems & Definitions (33)

  • Proposition 1.1
  • Theorem 1
  • Remark
  • Theorem 2
  • Remark
  • Theorem 3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • ...and 23 more