Shepp-Olkin 凹性问题中的尖锐 Rényi 和 Tsallis 阈值
The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem
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中文总结 AI 辅助
本研究精确确定了 Rényi 和 Tsallis 熵在 Shepp-Olkin 凹性问题中的联合凹性阈值，证明其通用范围恰为 $0<q\leq 1$，并给出了严格的数学证明。
中文摘要 AI 辅助
设 $B_1,\ldots,B_n$ 为参数分别为 $p_1,\ldots,p_n$ 的独立伯努利随机变量，令 $S=\sum_i B_i$。Hillion 和 Johnson 证明了 $S$ 的香农熵关于参数向量是联合凹的，并对 Rényi 和 Tsallis 熵提出了相应的临界阶猜想，预测阈值分别为 $2$ 和约 $3.65986$。我们精确确定了这两个阈值。\n对于每个 $0<q<1$，幂和 $\sum_k \mathbb P(S=k)^q$ 关于 $(p_1,\ldots,p_n)$ 是联合凹的，并且在开参数立方体上严格凹。因此，阶为 $q$ 的 Rényi 和 Tsallis 熵是联合凹的。在 $q=1$ 时，这与香农定理一致。对于每个 $q>1$，联合凹性在仅两个伯努利变量之和时就已经失效：两个参数反向移动的横向插值给出了两种熵的严格局部凸性。因此，两个族群的普遍联合凹性范围恰好是 $0<q\leq 1$。\n在阶数低于一的情况下，证明结合了 Hillion-Johnson 输运不等式与显式非线性望远镜修正。修正后的局部曲率归结为二维二次型。一个精确的 Riccati 恒等式，结合单侧过零论证，证明了其行列式在整个范围 $0<q<1$ 内为正。
英文摘要
Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\ldots,p_n$, and let $S=\sum_i B_i$. Hillion and Johnson proved that the Shannon entropy of $S$ is jointly concave in the parameter vector and proposed corresponding critical-order conjectures for R'enyi and Tsallis entropies, with predicted thresholds $2$ and approximately $3.65986$, respectively. We determine both thresholds exactly. For every $0<q<1$, the power sum $\sum_k \mathbb P(S=k)^q$ is jointly concave in $(p_1,\ldots,p_n)$, and strictly concave on the open parameter cube. Consequently, the R'enyi and Tsallis entropies of order $q$ are jointly concave. At $q=1$ this agrees with the Shannon theorem. For every $q>1$, joint concavity fails already for the sum of two Bernoulli variables: a transverse interpolation in which the two parameters move in opposite directions gives strict local convexity for both entropies. Hence the universal joint-concavity range for both families is exactly $0<q\leq 1$. Below order one, the proof combines the Hillion--Johnson transport inequality with an explicit nonlinear telescoping correction. The corrected local curvature reduces to a two-dimensional quadratic form. An exact Riccati identity, together with a one-sided zero-crossing argument, proves positivity of its determinant throughout the full range $0<q<1$.
