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arxiv: 2606.19313 · v2 · pith:RXM7HG2Snew · submitted 2026-06-17 · 🧮 math.PR

Stability of Khintchine-type inequalities via log-monotonicity

\'Angel Ch\'avez , Sam Sheng This is my paper

Pith reviewed 2026-06-26 19:36 UTC · model grok-4.3

classification 🧮 math.PR
keywords Khintchine inequalitieslog-monotonicitystability inequalitiesmoment ratiossymmetric random variablesweighted sumsL_p normsGaussian mixtures
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The pith

Log-monotonicity of the moment ratio sequence r_k(X) implies sharp L_p to L_2 norm comparisons for weighted sums of symmetric random variables at every even integer p at least 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that log-monotonicity of r_k(X) equals k factorial times E[X to the 2k] over (2k) factorial produces sharp comparisons between the L_p and L_2 norms of the weighted sum S of independent copies of a symmetric random variable X. This extends the classical Khintchine inequalities to all even p greater than or equal to 2 and supplies new bounds in the log-convex case. Two stability inequalities follow: one sharpens the bound by measuring deviation of the coefficient vector from its coordinate extremizers, and the other quantifies distance from the Gaussian limit. The results recover known stability statements for random signs and cover type-L variables, ultra sub-Gaussian variables, and Gaussian mixtures.

Core claim

If the sequence r_k(X) is log-monotonic, then for S equal to the sum of a_k X_k with independent symmetric copies X_k of X, the L_p norm of S is bounded above by an explicit constant times the L_2 norm for every even integer p at least 2, with the constant achieved at extremal coefficient vectors; the same monotonicity also yields quantitative stability versions that bound the defect from the extremal or Gaussian case.

What carries the argument

log-monotonicity of the sequence r_k(X) = k! E[X^{2k}] / (2k)!, which governs the growth of even moments and produces the sharp norm comparison constants

If this is right

  • For any log-monotonic X the inequality ||S||_p <= C_p(a) ||S||_2 holds with explicit C_p(a) for every even integer p >= 2.
  • The first stability inequality improves the bound proportionally to the distance of the coefficient vector a from the coordinate extremizers.
  • The second stability inequality bounds the defect from the Gaussian case in terms of a suitable distance between the law of X and the standard normal.
  • The same conclusions apply to type-L random variables, ultra sub-Gaussian random variables, and Gaussian mixtures.
  • The results recover the recent stability inequalities known for Rademacher random signs as a special case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same monotonicity condition could be checked directly for concrete families such as uniform or Laplace distributions to obtain explicit numerical bounds.
  • If a similar monotonicity property can be verified for odd moments or for nonsymmetric variables, the method would extend the inequalities beyond the even-p symmetric setting.
  • The stability forms might combine with concentration or small-ball estimates to produce tail bounds that improve when the coefficient vector is away from the worst case.

Load-bearing premise

The sequence r_k(X) must be log-monotonic; without this property the claimed sharp comparisons and stability statements need not hold.

What would settle it

Take a symmetric random variable X where the ratios log(r_{k+1}(X)/r_k(X)) are not nonincreasing, form the equal-coefficient sum S with four terms, and check whether E[|S|^4]^{1/4} / E[S^2]^{1/2} exceeds the constant that log-monotonicity would predict for p=4.

read the original 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 $r_k(X)=k! \mathbb{E}[X^{2k}]/(2k)!$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper shows that log-monotonicity of the sequence r_k(X) = k! E[X^{2k}]/(2k)! for a symmetric random variable X implies sharp comparisons between the L_p and L_2 norms of the weighted sum S = sum a_k X_k for every even integer p >= 2. It derives two stability inequalities, one sharpening the bound by the deviation of the coefficient vector from coordinate extremizers and the other quantifying deviation from the Gaussian limit. The results recover the random-sign case and are applied to type-L variables, ultra sub-Gaussian variables, and Gaussian mixtures.

Significance. If the derivations hold, the conditional framework via log-monotonicity supplies a unified route to sharp constants and stability statements that extends the classical Khintchine theory while remaining applicable to several concrete families. The explicit recovery of the Rademacher case and the breadth of the listed applications constitute concrete strengths.

minor comments (1)
  1. [Introduction / Applications section] The abstract states that the hypothesis is 'presumably verified separately' for the listed classes; the manuscript should make the verification steps for each class explicit in a dedicated subsection or appendix.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. Their summary accurately reflects the paper's contributions on log-monotonicity and its implications for Khintchine-type inequalities and stability results.

Circularity Check

0 steps flagged

No significant circularity; conditional implication from external assumption

full rationale

The paper's central claim is an implication: log-monotonicity of the explicitly defined sequence r_k(X) yields sharp L_p/L_2 comparisons for even p. This assumption is an independent property of X (verified separately for classes like type-L or Gaussian mixtures) and is not derived from or equivalent to the target inequalities. No self-citations, fitted parameters renamed as predictions, or self-definitional reductions appear in the abstract or described derivations. The results recover known cases without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the single domain assumption that r_k(X) is log-monotonic; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The sequence r_k(X) = k! E[X^{2k}] / (2k)! is log-monotonic for the symmetric random variable X under consideration.
    This property is invoked to derive the sharp norm comparisons and stability statements.

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