On a conjecture of Lamkin and Tkocz: Log-convexity of moments of Bernoulli sample means

Abstract: Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(θ)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence ${\mathsf{E}(\bar X_n^p)}_{n \geq 1}$ is log-convex. This settles the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison for a normalized quadratic function of a hypergeometric random variable. The log-convexity inequality is strict for $p &gt; 1$ and $0 < θ< 1$.

• The paper proves that for all p ≥ 1, the p-th moments of Bernoulli sample means are log-convex, with strict convexity for p > 1 and nontrivial parameters.
• It introduces a combinatorial-probabilistic reduction that embeds moment comparisons into hypergeometric frameworks to demonstrate convex-order dominance.
• The result has practical implications for tighter moment inequalities and improved finite-sample analysis in probability and statistical estimation.

Log-Convexity of Moments of Bernoulli Sample Means and a Solution to the Lamkin-Tkocz Conjecture

Introduction

The paper "On a conjecture of Lamkin and Tkocz: Log-convexity of moments of Bernoulli sample means" (2607.10545) resolves an outstanding conjecture concerning the log-convexity of the $p$-th moments of sample means of i.i.d. Bernoulli random variables. Specifically, for $X_1, X_2, \ldots$ i.i.d. $\operatorname{Bernoulli}(\theta)$ and $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$, the log-convexity of the sequence $E(\bar{X}_n^p)$ for every real $p \geq 1$ is established. The work completes the Bernoulli case of a conjecture originally formulated in [Lamkin and Tkocz, Canad. Math. Bull., 65(2):271–278, 2022], adding to existing results for other distributions.

Background and Problem Statement

Given a sequence of i.i.d. nonnegative random variables, the sequence $b_n(p) = E(\bar{X}_n^p)$ is central in analyzing concentration and dispersion properties of empirical means. While Jensen's inequality yields that $b_n(p)$ is non-increasing in $n$ for convex, non-decreasing $\varphi(x) = x^p$ ($X_1, X_2, \ldots$0), Lamkin and Tkocz conjectured a significantly stronger property: for all $X_1, X_2, \ldots$1, the sequence $X_1, X_2, \ldots$2 should be log-convex, i.e.,

$X_1, X_2, \ldots$3

Their conjecture was motivated by log-convexity/log-concavity phenomena observed in combinatorial and probabilistic sequences, but a full proof was only available for large $X_1, X_2, \ldots$4 and integer $X_1, X_2, \ldots$5 or for other distributions (such as Gamma and Poisson [Jiao & Li, Statist. Probab. Lett., 231, 110620, 2026]). The Bernoulli case, despite its foundational role in probability theory, remained unresolved and was singled out in prior work as a substantive open case.

Main Results

The main theorem in the paper demonstrates that for all $X_1, X_2, \ldots$6 and $X_1, X_2, \ldots$7, the sequence $X_1, X_2, \ldots$8 is log-convex. Furthermore, for $X_1, X_2, \ldots$9 and $\operatorname{Bernoulli}(\theta)$0, the log-convexity is {\bf strict}.

More precisely, letting

$\operatorname{Bernoulli}(\theta)$1

the result asserts that for all $\operatorname{Bernoulli}(\theta)$2,

$\operatorname{Bernoulli}(\theta)$3

whenever $\operatorname{Bernoulli}(\theta)$4 and $\operatorname{Bernoulli}(\theta)$5.

Proof Methods

A key novel element is a combinatorial-probabilistic reduction: both sides of the log-convexity inequality are embedded into distinct "splittings" over $\operatorname{Bernoulli}(\theta)$6 Bernoulli trials. Conditioning on the total number of successes $\operatorname{Bernoulli}(\theta)$7 restructures the analysis in terms of hypergeometric random variables $\operatorname{Bernoulli}(\theta)$8, representing the number of successes in respective subsets. This allows transformation of the required log-convexity property into a comparison of the $\operatorname{Bernoulli}(\theta)$9-th moments of two normalized quadratic functions of hypergeometric variables.

Formally, the proof shows that for every possible value of $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$0, the normalized quadratic forms satisfy a convex-order dominance:

$\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$1

where $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$2, and $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$3. The strictness for $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$4 is established by explicit evaluation for $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$5, exploiting the strict convexity of $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$6.

The argument invokes symmetry, interlacing properties of the supports, and telescoping identities for hypergeometric probability masses. The comparison in convex order proceeds via the stop-loss characterization, allowing a detailed analysis of all possible shifts $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$7.

Strong Claims and Numerical Properties

The most notable strong claim proved is the strict log-convexity for $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$8 and nontrivial $\bar{X}_n = n^{-1}\sum_{i=1}^n X_i$9, improving on the earlier understanding where only asymptotic or partial (integer $E(\bar{X}_n^p)$0, large $E(\bar{X}_n^p)$1) results were available. The work provides exact, non-asymptotic identities for $E(\bar{X}_n^p)$2, enabling immediate computation and verification for finite $E(\bar{X}_n^p)$3, $E(\bar{X}_n^p)$4, and $E(\bar{X}_n^p)$5.

Theoretical and Practical Implications

The resolution of this conjecture completes the canonical case for the family of i.i.d. Bernoulli means and solidifies the understanding of log-convexity phenomena in classical probability. Log-convexity of $E(\bar{X}_n^p)$6 underlies several inequalities in probability theory, with applications to concentration bounds, analysis of empirical mean estimators, and structural properties of functionals of sums of independent indicators.

From a theoretical perspective, the result motivates the search for similar exact convexity/concavity properties in the moments of more general statistics (e.g., U-statistics, empirical processes) and for additional distributions, potentially linking with combinatorial inequalities and stochastic orderings. The refined analysis of convex order dominance for quadratic functionals of hypergeometrics also has ramifications in optimal transport and coupling theory.

In applied settings (e.g., in statistics or information theory), tight control over the moments of averages is crucial in finite-sample analysis. The confirmed log-convexity delivers improved tail- and moment-type inequalities, relevant for risk management, algorithmic randomized methods, and robust estimation.

Future Directions

There are several immediate extensions: whether equivalent log-convexity properties extend to other symmetric or bounded distributions, the possibility of characterizing all distributions with this property, and refinement of the telescoping and symmetry methods to functionals beyond powers. There are potential links to statistical decision theory via convex risk measures and higher-order stochastic dominance.

In the context of AI, sharp understanding of moment structures of empirical means may have implications for the analysis of learning algorithms, concentration of measure in high-dimensional function spaces, and properties of noise in probabilistic models.

Conclusion

The paper (2607.10545) settles the Bernoulli case of the Lamkin-Tkocz conjecture by proving strict log-convexity of the moments $E(\bar{X}_n^p)$7 for all $E(\bar{X}_n^p)$8, completing a central structural property for sample mean moments of Bernoulli trials. The proof is technically innovative, leveraging new convex-order comparisons for normalized hypergeometric quadratic forms. The result enhances both theoretical understanding and practical bounds in probabilistic analysis of averages, and suggests fruitful avenues for deeper inquiry into log-convexity in stochastic processes.