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Fourth-Moment Geometry of Rademacher Sums

Peigan Gao, Jian Qian

arXiv:2608.17802v1cs.LGmath.PR

TL;DR

The paper studies how higher moments of normalized Rademacher sums depend on fourth-order mass, addressing concentration and fixed-dimensional optimizer geometry. It develops fixed-q extremal and convexity arguments to prove Gaussian stability for p≥4, sharp finite-dimensional L_p/L_4 constants for p≥5, and a quadratic stability estimate at p=3.

  • Problem

    Classical Khintchine bounds suppress concentration behavior and fixed-dimensional optimizer geometry, motivating fourth-order mass as a shared parameter.

  • Method

    The proof uses an extremal reduction with at most one exceptional coefficient, fixed-q convexity, and a separate averaging argument for the critical third moment.

  • Results

    The paper proves sharp Gaussian stability for p≥4, a fixed-fourth-moment extremal principle, and sharp finite-dimensional L_p/L_4 constants for p≥5.

  • Takeaways & Limitations

    The extremal principle yields coefficient-sensitive Laplace-transform and tail bounds that distinguish vectors with equal variance but different effective support sizes.

  • Takeaways & Limitations

    The chord inequality fails for every 2<p<4, marking a scope boundary for this linear-in-q estimate.

Abstract

from arXiv · show

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher signs and let $a=(a_1,\ldots,a_n)\in\R^n$ satisfy the normalization below. For the normalized Rademacher sum, we determine how its higher moments depend on the fourth-order mass. Combining a sharp fixed-q moment envelope with a separate argument below the convexity threshold gives the Gaussian stability inequality for the full range $p\geq4$ of this linear-in-q bound. The same fourth-order framework determines the sharp finite dimensional $L_p/L_4$ Khintchine constant for $p\geq5$, with the flat coefficient vector as the extremizer. These results settle the conjectures of Jakimiuk and of Barański, Murawski, Nayar, and Oleszkiewicz stated below. We also prove Jakimiuk's conjectured quadratic stability estimate at $p=3$. The resulting bounds retain information about sparsity and effective dimension, with applications to Rademacher random projections and randomly signed errors; those applications are not developed further here. Their Laplace-transform form also gives coefficient-sensitive tail bounds. The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol.

1 Introduction

The paper uses fourth-order mass q to track both non-Gaussianity and coefficient concentration in normalized Rademacher sums. It proves sharp Gaussian-stability, fixed-q, finite-dimensional, third-moment, and concentration results, resolving the stated conjectures.

  • Fourth-order framework: The fourth-order mass q measures the fourth-moment defect from Gaussianity and coefficient concentration, with q−1 serving as an effective number of active coordinates.The parameter ranges from q = 1 for a coordinate vector to q = 1/N for a flat vector in dimension N.
  • Sharp Gaussian stability: The sharp Gaussian stability theorem establishes the optimal linear-in-q coefficient for the full range p ≥ 4, with equality for every coefficient vector at p = 4 and only coordinate vectors for p > 4.The latter equality characterization is up to signs and permutations.
  • Fixed-fourth-moment extremal principle: For convex fourth derivatives, the fixed-fourth-moment principle identifies the extremal law at prescribed q and proves its envelope is strictly convex and strictly below the Gaussian-to-coordinate chord in the interior.The supremum is generally attained only in the closure of finite Rademacher sums, through one macroscopic coefficient and increasingly many vanishing coefficients.
  • Finite-dimensional Lp/L4 constant: For every real p ≥ 5 and N ≥ 2, the finite-dimensional Lp/L4 extremizer is the flat coefficient vector, unique after normalization up to coordinate signs and permutations.Without normalization, uniqueness also allows nonzero scalar multiplication.
  • Dimension-free stability at the third moment: The paper proves a dimension-free quadratic stability estimate at the third moment, including the sharp endpoint description in dimension three and the optimizer (1/2, 1/2, 0) up to signs and permutations.This confirms Jakimiuk’s second conjecture and identifies the endpoint where the usual smooth-convexity argument degenerates.
  • Coefficient-sensitive concentration: The fixed-q principle yields optimal coefficient-sensitive Laplace-transform and tail bounds that interpolate between the Gaussian regime q → 0 and a single random sign at q = 1.The extremal moment generating function is that of the one-spike-plus-Gaussian law Yq.

2 A direct proof of sharp Gaussian stability for 4 ≤p ≤6

The proof establishes sharp Gaussian stability for 4 ≤ p ≤ 6 by splitting at q = 1/2: coefficientwise estimates handle small q, while fourth–sixth moment interpolation handles large q. Combining the two propositions proves the target inequality throughout the full range.

  • 2 A direct proof of sharp Gaussian stability for 4 ≤p ≤6: q = 1/2 separates the proof into coefficientwise control for q ≤ 1/2 and fourth–sixth moment interpolation for q ≥ 1/2.The small-q argument uses a uniform coefficientwise bound, while the large-q argument interpolates between the exact fourth moment and an upper sixth-moment bound.
  • 2 A direct proof of sharp Gaussian stability for 4 ≤p ≤6: For q ≤ 1/2, the coefficientwise estimate is reduced to a bound in q using monotonicity of ϕp and the condition maxi xi ≤ 2^-1/2.The maximum-coordinate bound follows from q ≤ 1/2 and (maxi xi)^2 ≤ q.
  • 2 A direct proof of sharp Gaussian stability for 4 ≤p ≤6: The coefficientwise inequality is strict for p > 4.The strictness follows from the monotonicity argument used to express the estimate solely in terms of q.
  • 2 A direct proof of sharp Gaussian stability for 4 ≤p ≤6: For q ≥ 1/2, Proposition 2.5 proves Equation (3) for 4 ≤ p ≤ 6.Its proof uses log-convexity between the fourth and sixth moments together with ES4 = 3 − 2q.
  • 2 A direct proof of sharp Gaussian stability for 4 ≤p ≤6: Combining Proposition 2.3 for q ≤ 1/2 with Proposition 2.5 for q ≥ 1/2 proves Equation (3).The two q-ranges cover the full normalized parameter range used in the proof.

3 The fixed-moment reduction for p ≥5

For p ≥5, the extremal problem reduces to one distinguished coefficient and equal smaller coefficients, whose infinite-dimensional limit is a Gaussian shift. A Gaussian-shift chord inequality then yields the target inequality for all p ≥5, completing the full p ≥4 range together with the earlier result.

  • Gaussian-shift limit: Letting the number of equal smaller coefficients tend to infinity produces a Gaussian shift, reducing the remaining comparison to one dimension.The convergence uses the Lindeberg–Feller central limit theorem and uniform integrability of the p-th moments.
  • Extremal reduction: For p ≥5, Lemma 3.1 reduces the maximizer to a coefficient vector of the form (c, d, …, d), with c ≥ d ≥ 0.The reduction fixes the second- and fourth-moment constraints.
  • Gaussian-shift comparison: For p > 4, the Gaussian-shift profile lies below its endpoint chord, with strict inequality for 0 < u < 1.The profile has endpoint values g(0) = µp and g(1) = 1, and its strict convexity gives the chord inequality.
  • Main consequence: The inequality Equation (3) holds for every p ≥5.Proposition 3.4 follows by combining the fixed-moment reduction with the Gaussian-shift chord inequality at u = q1/4.
  • Completion of Theorem 1.1: Theorem 1.1 holds for the full range p ≥4, and coordinate vectors give equality for every p > 4.Corollary 2.6 covers 4 ≤ p ≤ 6, while Proposition 3.4 covers p ≥5; together they cover p ≥4.

4 The exact fixed-q upper envelope

The fixed-moment principle gives the exact finite-dimensional upper envelope at each feasible fourth moment q, with extremizers reduced to one distinguished coefficient and equal remaining coefficients. It also yields sharp fixed-q Laplace transforms, strict equality characterization for p>4, and failure of the conjectured inequality below p=4.

  • Exact finite-dimensional optimization: Proposition 4.1 identifies the exact fixed-q supremum for even Φ with convex fourth derivative, reducing maximizers to coefficients of the form (c, d, ..., d).The parameters satisfy c^2 + (N − 1)d^2 = 1, c^4 + (N − 1)d^4 = q, and c ≥ d ≥ 0.
  • Exact finite-dimensional optimization: The upper envelope is the exact supremum over finite Rademacher sums, approached after appending zeros and passing to the finite-sum closure.The limiting fourth-moment parameter is feasible whenever Nq ≥ 1, and expectations converge through uniform integrability.
  • Moment-profile and equality cases: For p > 4, equality holds only at q = 1, equivalently when exactly one coefficient is nonzero and has magnitude 1; at p = 4, every coefficient vector gives equality.For p ≥ 5, strict convexity of q ↦ U_p(q) gives strict inequality for 0 < q < 1, while the argument extends strictness to 4 < p < 5.
  • Moment-profile and equality cases: For every 2 < p < 4, the conjectured inequality fails for S_2 = (ε_1 + ε_2)/√2, with E|S_2|^p > μ_p + 1.This example has q = 1/2 and E|S_2|^p = 2^(p/2−1).
  • Sharp fixed-q Laplace transform: The fixed-q principle also determines the sharp Laplace-transform supremum, attained as an exact supremum in the finite-sum closure.The result applies to E exp(tΣa_iε_i), using Φ(x) = cosh(tx) and symmetry of S.

5 The finite-dimensional Lp–L4 conjecture

For p ≥ 5, the paper proves the conjectured flat-vector extremality for the finite-dimensional Lp/L4 Khintchine ratio by establishing a stronger strict monotonicity statement. The result also characterizes the optimizer uniquely up to coordinate signs and permutations.

  • The finite-dimensional Lp–L4 conjecture: For p ≥ 5, the proof applies the cubic-quotient lemma after showing the relevant third derivative is even and convex.A Rademacher moment recurrence verifies the lemma’s strict initial condition, producing strict increase of the auxiliary quotient.
  • The finite-dimensional Lp–L4 conjecture: The ratio is strictly decreasing on (1, ∞), with the n = 1 case following separately from the quotient cp−4.For n = 1, cp−4 is strictly increasing for p > 4, yielding the same ratio decrease.
  • The finite-dimensional Lp–L4 conjecture: The flat coefficient vector is the unique normalized optimizer up to coordinate signs and permutations.Without normalization, the nonzero scalar multiples of the flat vector are also optimizers.

6 Dimension-free stability at the critical third moment

At p = 3, a squared-coefficient averaging argument combined with a small-ball estimate yields dimension-free stability toward the flat vector. The proof also uses a sharp dimension-three inequality, while the exact stability constant remains undetermined.

  • Cubic smoothing: At p = 3, averaging the largest and smallest squared coefficients preserves their sum, moves toward the flat point, and decreases q.Conditioning on the remaining signs reduces the third-moment gain to a two-variable estimate, made uniform by a small-ball bound.
  • Cubic smoothing: The cubic two-point smoothing lemma shows that replacing an extreme pair by its average produces a nonnegative conditional gain in the third moment.Averaging the conditional estimate over the remaining coordinates gives the extreme-pair gain used in the global iteration.
  • Dimension-three inequality: In dimension three, the sharp quotient is attained by the squared-coefficient vector (1/2, 1/2, 0), up to signs and permutations.The proof separates the regions a ≥ b + c and a ≤ b + c and identifies equality at the non-flat endpoint.
  • Global iteration: Iterating extreme-pair averaging drives the squared-coefficient vector to the flat point and, together with the preceding inequality, proves the dimension-free stability bound.The argument uses convergence of the averaging sequence and continuity of the moment functional.
  • Limitations: The proof establishes dimension-free stability but does not determine the exact value of the stability constant.The dimension-three vector is identified as a natural sharpness candidate, and improving the small-ball constant 3/16 would improve the method’s lower bound.

7 Concluding perspective … B Calculations for the third-moment argument

The concluding perspective identifies two extremal geometries governed by fourth-power mass: a spike-plus-Gaussian-cloud law at fixed q and a one-spike perturbation of flat coefficients at fixed dimension. The appendices provide curvature, comparison, and endpoint calculations supporting the Gaussian-shift argument.

  • 7 Concluding perspective: At fixed q, the upper law combines one Bernoulli spike with a Gaussian cloud, whose endpoint comparisons yield Gaussian stability.The fourth-power mass q exposes this extremal geometry.
  • 7 Concluding perspective: At fixed dimension, the fourth-order principle permits one spike among otherwise equal coefficients, while the cubic-quotient lemma makes the flat choice optimal.The interval 4 < p < 5 falls below the fixed-moment principle’s convexity threshold and requires a separate argument.
  • A Calculations for the Gaussian-shift chord: The Gaussian-shift appendix supplies the curvature and comparison computations used in Lemma 3.3.These calculations organize the supporting argument for the main result.
  • A.1 The curvature calculation: Gaussian integration by parts, repeated applications of Equation (58), and Equation (22) produce the curvature identities needed for the argument.The appendix records the differentiation steps leading to Equation (23).
  • A.2 The Riccati comparison: The Riccati comparison is established by reducing the relevant expressions to a common denominator and proving Equation (26).Equation (22) supplies Equation (25), while the direct comparison proves Equation (26).
  • A.2 The Riccati comparison: At the singular endpoint t = 0, variation of constants applies because E(r) = O(r4) and I(r) = O(r), and positivity of the integrand gives P(t) > 0 for every t > 0.Here P = Q − R, and the endpoint condition is P(t) → 0 as t ↓ 0.

B.1 The two-point profile

This section verifies the monotonicity required in Lemma 6.2 for a normalized two-point profile. The argument shows that G is nonincreasing on [0,1] and establishes the remaining nonnegativity assertion.

  • Monotonicity: G is nonincreasing on [0,u], and therefore on [0,1], based on the two branches of H and concavity of the left-hand side.The endpoint values of the concave left-hand side are at least 2.
  • Setup: The verification uses normalized variables satisfying x + y = 1 and u = √x + √y, v = √x −√y, with u2 + v2 = 2.
  • Nonnegativity: When all terms lie in the second branch of H, direct substitution gives G(s) = 0, proving the remaining nonnegativity assertion used in the main proof.

B.2 The dimension-three reduction · C The scalar estimates

The section supplies two calculations for the dimension-three reduction, establishing the needed scalar inequality through ordered feasibility, minimization, and monotonicity. It also records elementary scalar estimates used in Lemmas 2.2 and 2.4 without numerical optimization.

  • B.2 The dimension-three reduction: C⋆=5 is introduced in the calculations used to prove Lemma 6.4.
  • B.2 The dimension-three reduction: The reduction fixes s=a+b+c, orders a≥b≥c≥0, and imposes a^2+b^2+c^2=1.
  • B.2 The dimension-three reduction: The ordered feasible interval for a begins at the boundary a=b.
  • B.2 The dimension-three reduction: Consequently, abc is minimized at a=b.
  • B.2 The dimension-three reduction: For Equation (53), differentiating its left-hand side J(z) reduces the argument to the behavior of a derivative factor.
  • B.2 The dimension-three reduction: The derivative factor is strictly increasing on [0,1].
  • B.2 The dimension-three reduction: Because the factor is negative at z=0 and positive at z=1, J first decreases and then increases, so it has no interior maximum and Equation (53) follows.
  • C The scalar estimates: The appendix records elementary estimates used in Lemmas 2.2 and 2.4, with no numerical optimization involved.

C.1 The estimate for small fourth-power mass

The section proves the desired inequality by showing D(4)=0 and D′(p)>0 for every p≥4. Consequently, D(p)≥0, with strict inequality for p>4.

  • Monotonicity argument: D(4)=0 after substituting µ4=3, so proving the inequality reduces to showing that D is strictly increasing.The target inequality is equivalent to D(p)≥0.
  • Conclusion: D(p)≥0 for p≥4, and the inequality is strict when p>4.This follows from D(4)=0 together with the strict positivity of D′(p).
  • Monotonicity argument: D′(p)>0 is reduced to proving that a ratio exceeds one, using positivity of the relevant logarithmic terms.Both −ρ(p−2)/2 log ρ/(2x0) and ℓ(p)/µp are positive.
  • Monotonicity argument: R is strictly increasing and satisfies R(4)>1, so R(p)>1 for every p≥4 and therefore D′(p)>0.The proof uses bounds including ρ>e−2/3 and estimates derived from γE+log 2>19/15.

C.2 The estimate for large fourth-power mass · C.3 Elementary constants

The large-fourth-power-mass estimate is established through a logarithmic inequality, convexity, and a supporting-tangent argument. Elementary bounds on constants justify the argument, while the split at q = 1/2 is intentionally unoptimized.

  • C.2 The estimate for large fourth-power mass: The proof begins with a logarithmic estimate for q ∈ [1/2, 1], using Equation (63) and elementary bounds on γE and log 2.These bounds imply ℓ(4) > log 2.
  • C.2 The estimate for large fourth-power mass: B(x) ≥ 1 on 2^-1/2 ≤ x ≤ 1 because B′(x) = 12x(4x − 5) < 0 and B(1) = 1.This ensures that the logarithm used in the estimate is well defined.
  • C.2 The estimate for large fourth-power mass: N is strictly convex because N′′′(x) < 0, N′′ decreases, and N′′(1) = 136 − 108 log 2 > 0.The resulting monotonicity controls the number and location of zeros relevant to the argument.
  • C.2 The estimate for large fourth-power mass: Endpoint signs and increasing N′ force exactly one zero of N on its decreasing branch, after which K first increases and then decreases.Using log 2 < 7/10 yields K ≥ 0, which is Equation (64).
  • C.2 The estimate for large fourth-power mass: For q ∈ [1/2, 1], convexity of p 7→log Λp(q) and the tangent at p = 4 combine with Equation (64) to produce the desired estimate.Here Λp(q) is the p-th moment of a variable equal to 1 with probability q and |G| otherwise.
  • C.3 Elementary constants: Lemma C.1 supplies elementary estimates for log 2, γE, and related constants using power-series truncation, geometric tails, and Euler–Maclaurin inequalities.The proof also derives log 7 < 109/56 and γE > 23/40.
  • C.3 Elementary constants: The split at q = 1/2 is deliberately unoptimized because the two scalar arguments have transparent endpoint checks and meet without numerical optimization.This explains the chosen threshold rather than claiming it is optimal.
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