Stability of Khintchine-type inequalities via log-monotonicity

Abstract: We investigate Khintchine-type inequalities for the weighted sums $S=\sum_ka_kX_k$ of independent copies of a symmetric random variable $X$. We show how log-monotonicity of the sequence $r_k(X)=k! \mathbb{E}[X<sup>{2k}]/(2k)!$ implies sharp comparisons between the $L_p$ and $L_2$ norms of $S$ for every even integer $p\geq 2$, extending classic Khintchine-type inequalities and yielding new results in the log-convex setting. We also investigate the stability of our inequalities. Our first stability inequality sharpens the classic inequality by a deviation of the coefficient vector from the coordinate extremizers, while the second quantifies deviation from the Gaussian limit. Our results recover recent stability inequalities for random signs and apply to a broad class of distributions, including type-$\mathscr{L}$ random variables, ultra sub-Gaussian random variables and Gaussian mixtures.

• The paper develops a framework using log-monotonicity to extend classical Khintchine inequalities for weighted sums of symmetric random variables.
• It introduces sharp two-sided Lp norm bounds and stability defect terms that quantify deviations from both coordinate and Gaussian extremizers.
• The results apply to diverse distributions like ultra sub-Gaussian and Gaussian mixtures, enhancing the precision of moment comparisons.

Stability of Khintchine-type Inequalities via Log-Monotonicity

Overview

The paper "Stability of Khintchine-type inequalities via log-monotonicity" (2606.19313) develops a refined analysis of Khintchine-type inequalities for weighted sums of independent symmetric random variables $X$, focusing on the comparison between $L_p$ and $L_2$ norms for even $p \geq 2$. The central innovation is the introduction of log-monotonicity (log-concavity/log-convexity) for sequences of normalized moments, $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$, providing a unified framework to extend classical Khintchine-type inequalities and sharpen their stability properties. The results encompass classical cases, ultra sub-Gaussian distributions, type-$\mathscr{L}$ random variables, Gaussian mixtures, and offer new bounds for log-convex scenarios.

Log-Monotonicity and Schur-Monotonicity

The classical Khintchine inequality quantifies the relationship between the $L_p$ and $L_2$ norms of sums $S = \sum_k a_k X_k$, originally for Rademacher random variables. The paper generalizes this context by considering $X$ with symmetric distributions and focusing on the normalized moment sequence $L_p$0. Central to the framework is log-monotonicity:

• Log-concavity: $L_p$1
• Log-convexity: $L_p$2

These conditions imply Schur-monotonicity properties for distributions:

• If $L_p$3 is log-concave, $L_p$4 is Schur-concave for even $L_p$5.
• If $L_p$6 is log-convex, $L_p$7 is Schur-convex for even $L_p$8.

This grants explicit inequalities for $L_p$9 norms, establishing sharp two-sided bounds parameterized by the distribution's moment structure.

Main Results and Stability

Sharp Two-Sided Inequalities

Let $L_2$0, $L_2$1, and $L_2$2 iid copies of a symmetric $L_2$3. For $L_2$4 and even $L_2$5, the following holds:

• Log-concave case: $L_2$6
• Log-convex case: $L_2$7

where $L_2$8 is the $L_2$9 norm of a standard Gaussian, and optimality is established.

Stability Enhancements

To quantify deviation from extremal configurations, two stability results are presented.

Coordinate Stability

For $p \geq 2$0 and log-monotone $p \geq 2$1:

• Log-concave: $p \geq 2$2
• Log-convex: $p \geq 2$3

These strengthen the classic bounds by introducing defect terms measuring the distance from coordinate extremizers.

Gaussian Stability

For $p \geq 2$4:

• Log-concave: $p \geq 2$5
• Log-convex: $p \geq 2$6

where $p \geq 2$7 quantifies deviation from the Gaussian extremizer, and is explicitly computable. The constants are asymptotically sharp near extremal vectors.

Extension to Broader Classes

The framework applies to ultra sub-Gaussian random variables (by Nayar and Oleszkiewicz) and type-$p \geq 2$8 variables (per Newman and Havrilla et al.), confirming that these classes satisfy the Schur-concave property when $p \geq 2$9 is log-concave. The paper also explores distributions with log-convex moment sequences, notably Gaussian mixtures and certain exponential functionals.

Implications and Open Problems

Numerical Results and Contradictory Claims

The constants in the stability terms are rigorously proven to be sharp, both near coordinate and Gaussian extremizers. The paper proves that the lower bound in the log-concave case and the entirety of the log-convex case are new; prior methods (e.g., those in Havrilla et al., Nayar, Newman) do not apply to distributions with log-convex $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$0, highlighting the novelty of these results.

The paper demonstrates, for instance, that for Rademacher variables, its Gaussian stability theorem recovers and refines Jakimiuk's recent stability inequalities, yielding explicit constants such as $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$1 for $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$2, a strong numerical assertion.

Practical and Theoretical Implications

The established bounds and stability terms are uniform for a wide array of symmetric distributions, enabling robust moment comparison tools for applications in probability, functional analysis, and random matrix theory. For distributions beyond Rademacher (e.g., spherical, ultra sub-Gaussian, type-$r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$3, Gaussian mixtures), practitioners now possess explicit control over $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$4 norm deviations, relevant for concentration inequalities, isoperimetric estimates, and stability in randomized constructions.

The precision and generality facilitate further analysis in settings where the central limit behavior or extremal properties of moments govern the outcome, and provide templates for the investigation of higher-dimensional or dependent random variable structures.

Future Directions

Several conjectures and open problems are posed. Notably, the "Diagonal Stability Conjecture" proposes explicit defect bounds measuring the distance from uniform extremizer for general log-monotone distributions, paralleling established results for Rademacher variables. The extension of Schur-monotonicity to further classes of symmetric random variables with non-log-monotone moment sequences remains unresolved. Identifying distributions with the Schur-monotone property but without log-monotonicity is an open problem.

Anticipated developments include sharpening constants for wider regimes, identifying broader classes with log-convex behavior, and establishing analogs for odd $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$5 or non-symmetric $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$6.

Conclusion

By integrating log-monotonicity of normalized moment sequences with Schur-monotonicity and stability considerations, this paper generalizes and strengthens Khintchine-type inequalities for symmetric random variables. It introduces sharp, explicit bounds for the $r_k(X) = k! \mathbb{E}[X^{2k}] / (2k)!$7 norm of weighted sums, quantifies the defect terms measuring departure from extremizers, and extends the analysis to both log-concave and log-convex cases. The results unify prior advances, provide new inequalities and stability measures for important classes beyond Rademacher, and set forth conjectures for future exploration. The framework contributes both theoretical depth and practical tools for researchers in probability, functional analysis, and related areas.