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Distributional stability of the Szarek and Ball inequalities

Alexandros Eskenazis, Piotr Nayar, Tomasz Tkocz

TL;DR

This work establishes distributional stability for sharp Khinchin-type inequalities under perturbations of the underlying law. Using Haagerup's Fourier-analytic method, it proves that when i.i.d. symmetric variables are within a small $W_2$ distance of the Rademacher distribution and the weight vector satisfies $\|a\|_\infty \le 1/\sqrt{2}$, the Szarek-type inequality $\mathbb{E}|\sum a_j X_j| \ge \mathbb{E}|(X_1+X_2)/\sqrt{2}|$ persists up to a universal tolerance. An analogous result is obtained for Ball's cube-slicing inequality in $\mathbb{R}^3$ under decay of the characteristic function and a stringent $W_2$ proximity to the sphere distribution. The analysis hinges on bounds and monotonicity properties of two special functions, $\Psi$ and $\Phi$, which compare the perturbed distributions to the canonical Rademacher and sphere models, respectively. The results illustrate robustness of these optimal inequalities to distributional perturbations and open avenues for extending sharp probabilistic inequalities to broader distributional classes. Participation of Fourier-analytic representations and precise control of error terms makes the findings applicable to further stability questions in high-dimensional probability and convex geometry.

Abstract

We prove an extension of Szarek's optimal Khinchin inequality (1976) for distributions close to the Rademacher one, when all the weights are uniformly bounded by a $1/\sqrt2$ fraction of their total $\ell_2$-mass. We also show a similar extension of the probabilistic formulation of Ball's cube slicing inequality (1986). These results establish the distributional stability of these optimal Khinchin-type inequalities. The underpinning to such estimates is the Fourier-analytic approach going back to Haagerup (1981).

Distributional stability of the Szarek and Ball inequalities

TL;DR

This work establishes distributional stability for sharp Khinchin-type inequalities under perturbations of the underlying law. Using Haagerup's Fourier-analytic method, it proves that when i.i.d. symmetric variables are within a small distance of the Rademacher distribution and the weight vector satisfies , the Szarek-type inequality persists up to a universal tolerance. An analogous result is obtained for Ball's cube-slicing inequality in under decay of the characteristic function and a stringent proximity to the sphere distribution. The analysis hinges on bounds and monotonicity properties of two special functions, and , which compare the perturbed distributions to the canonical Rademacher and sphere models, respectively. The results illustrate robustness of these optimal inequalities to distributional perturbations and open avenues for extending sharp probabilistic inequalities to broader distributional classes. Participation of Fourier-analytic representations and precise control of error terms makes the findings applicable to further stability questions in high-dimensional probability and convex geometry.

Abstract

We prove an extension of Szarek's optimal Khinchin inequality (1976) for distributions close to the Rademacher one, when all the weights are uniformly bounded by a fraction of their total -mass. We also show a similar extension of the probabilistic formulation of Ball's cube slicing inequality (1986). These results establish the distributional stability of these optimal Khinchin-type inequalities. The underpinning to such estimates is the Fourier-analytic approach going back to Haagerup (1981).
Paper Structure (15 sections, 20 theorems, 118 equations)

This paper contains 15 sections, 20 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['thm:mainS']}
  4. A uniform bound on the characteristic function
  5. Uniform bounds on the special function and its derivative
  6. Proof of Lemma \ref{['lm:Psi2']}
  7. Proof of Theorem \ref{['thm:mainB']}
  8. A uniform bound on the characteristic function
  9. Bounds on the special function
  10. The bulk
  11. The Gaussian approximation
  12. Bounds on the derivative of the special function
  13. Bounds on Ball's special function
  14. Proof of Lemma \ref{['lm:Phi2']}
  15. Concluding remarks

Key Result

Theorem 1

There is a positive universal constant $\delta_0$ such that if we let $X_1, X_2, \dots$ be i.i.d. symmetric random variables satisfying then for every $n \geq 3$ and unit vectors $a$ in $\mathbb{R}^n$ with $\|a\|_\infty \leq \frac{1}{\sqrt{2}}$, we have Moreover, we can take $\delta_0 = 10^{-4}$.

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 30 more