AI 中文总结
研究伯努利样本均值矩的对数凸性，通过基于\(2n\)次试验中成功总数的证明方法，将不等式简化为超几何随机变量相关的凸序比较，解决了拉姆金和托茨猜想的伯努利情形。
AI 中文摘要
设\(X_1,X_2,\ldots\)为独立的\(\mathrm{Bernoulli}(\theta)\)随机变量，\(\bar X_n = n^{-1}(X_1 + \cdots + X_n)\)。我们证明，对于每个实数\(p \geq 1\)，序列\(\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}\)是对数凸的。这解决了拉姆金和托茨[《加拿大数学通报》，65(2):271 - 278，2022]猜想的伯努利情形。证明基于\(2n\)次试验中成功的总数，并将所需不等式简化为超几何随机变量的归一化二次函数的凸序比较。对于\(p > 1\)和\(0 < \theta < 1\)，对数凸性不等式是严格的。
英文摘要
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 proves 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 between two normalized quadratic functions of hypergeometric random variables. All the log-convexity inequalities are strict for $p > 1$ and $0 < θ< 1$.
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