Let p≥ 2 be an even integer and let (a_1,a_2,…,a_n)∈R^n satisfy ∑_k a_k^2=1. Let X be a symmetric random variable, let X_1,X_2,…,X_n be independent identically distributed copies of X, and write r_k=r_k(X). Define S=∑_k a_kX_k,qquad Y_n=frac1√ n(X_1+X_2+…+X_n). Diagonal stability conjecture. There exist positive constants C_p>0 such that, if (r_k) is log-concave, then |S|_p^p≤ |Y_n|_p^p-C_p∑_k(a_k^2-frac1n)^2, and, if (r_k) is log-convex, then |S|_p^p≥ |Y_n|_p^p+C_p∑_k(a_k^2-frac1n)^2. This conjecture proposes a quantitative stability estimate for the diagonal case of Schur-type Khintchine inequalities: the gap from the equal-weight vector is controlled by the squared deviations of the coefficients from 1/n. The paper leaves it as an open problem.

Diagonal stability conjecture for Khintchine-type inequalities

From papers

Let $p\geq 2$ be an even integer and let $(a_1,a_2,\ldots,a_n)\in\mathbb{R}^n$ satisfy

\sum_k a_k^2=1.

Let $X$ be a symmetric random variable, let $X_1,X_2,\ldots,X_n$ be independent identically distributed copies of $X$ , and write $r_k=r_k(X)$ . Define

S=\sum_k a_kX_k,\qquad Y_n=\frac{1}{\sqrt n}(X_1+X_2+\cdots+X_n).

Diagonal stability conjecture. There exist positive constants $C_p>0$ such that, if $(r_k)$ is log-concave, then

\|S\|_p^p\leq \|Y_n\|_p^p-C_p\sum_k\left(a_k^2-\frac{1}{n}\right)^2,

and, if $(r_k)$ is log-convex, then

\|S\|_p^p\geq \|Y_n\|_p^p+C_p\sum_k\left(a_k^2-\frac{1}{n}\right)^2.

This conjecture proposes a quantitative stability estimate for the diagonal case of Schur-type Khintchine inequalities: the gap from the equal-weight vector is controlled by the squared deviations of the coefficients from $1/n$ . The paper leaves it as an open problem.

Progress summary

Open

A 2026 paper records the conjecture as open; only a special random-sign case is known.

The conjecture asks for a quantitative gap from equal coefficients under log-concavity or log-convexity of the moment sequence. Chávez and Sheng state it as Conjecture 7.1 and explicitly leave it open.

Known results

Rademacher variables, every $p\geq 4$ : Jakimiuk proved the corresponding diagonal stability inequality with a positive constant and the squared-deviation term.

June 2026 status report

Chávez and Sheng’s paper, published June 17, 2026, formulates the conjecture and records the Rademacher special case, but reports no proof for general symmetric random variables and no counterexample or claimed settlement.

Current status (as of August 2026): The conjecture remains open for general symmetric random variables; the only recorded positive result is the Rademacher case for $p\geq 4$ .

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Ángel Chávez and Sam Sheng, “Stability of Khintchine-type inequalities via log-monotonicity”, arXiv:2606.19313 (2026).

1 week ago

Both proposed diagonal-stability inequalities are false for every even p≥2, even for a smooth, nondegenerate, unit-variance Gaussian distribution.

The source defines

r_j(X)=j! E[X^{2j}]/(2j)!.

Take X=G∼N(0,1), n=2, and (a₁,a₂)=(1,0). Since

E[G^{2j}]=(2j)!/(2^j j!),

we have

r_j(G)=2^{−j},

which is simultaneously log-concave and log-convex:

r_j(G)²=r_{j−1}(G)r_{j+1}(G).

Thus both stated hypotheses apply. But

S=G₁, Y₂=(G₁+G₂)/√2

have the same standard Gaussian distribution, while

(a₁²−1/2)²+(a₂²−1/2)²=1/2.

Writing m_p=E|G|^p, the two proposed conclusions become respectively

m_p≤m_p−C_p/2,

m_p≥m_p+C_p/2.

Both are impossible for every C_p>0 and every even p≥2. This remains false even if C_p is permitted to depend on the distribution.

There is an independent universal endpoint obstruction. For every symmetric finite-variance X and every normalized coefficient vector,

E|S|²=E[X²]=E|Y_n|².

Hence at the included exponent p=2, any nonuniform squared-weight vector contradicts either claimed positive deficit, irrespective of log-monotonicity.

At p=4 there is also a sharp corrected theorem. Put μ₂=E[X²], μ₄=E[X⁴], and

D(a)=∑_{i=1}^n(a_i²−1/n)²=∑_i a_i⁴−1/n.

Independence gives the exact identity

E[S⁴]−E[Y_n⁴]=(μ₄−3μ₂²)D(a).

Because r₀=1, r₁=μ₂/2, and r₂=μ₄/12, log-concavity implies μ₄≤3μ₂² and log-convexity implies μ₄≥3μ₂². The corresponding optimal fixed-distribution stability constants are therefore

C₄=3μ₂²−μ₄ in the log-concave case,

C₄=μ₄−3μ₂² in the log-convex case.

They are strictly positive exactly when the relevant kurtosis inequality is strict. For the Gaussian, both vanish, explaining precisely why the weak log-concavity/log-convexity assumptions in Conjecture 7.1 cannot imply a positive stability constant.

0 endorsements

Shivam Patel · 21 hours ago