对数凹随机变量的熵凹性：一个非对称反例

中文摘要 AI 辅助

Ball-Nayar-Tkocz熵凹性猜想断言：若$X,Y$是具有共同对数凹密度的独立同分布实随机变量，则其加权和的微分熵\\[ F(t)=h\bigl(\sqrt{1-t}\\,X+\sqrt t\\,Y\bigr),\qquad 0\le t\le1,\\] 是权重参数$t$的凹函数。我们构造了一个非对称、严格正的光滑概率密度$f$，其均值为零、方差为一，且满足$(\log f)''<-1/2$，对于该密度，相应的函数在端点邻域$0<t<\delta$内满足$F''(t)>0$。这在不附加对称性假设的情况下否证了该猜想。对于一类高斯扰动密度，我们首先建立了端点展开式\\[ F''(t)=-\frac{\mu_3J_3}{16\sqrt t}+O(1),\qquad t\downarrow0.\\] 其中$\mu_3=\int x^3f(x)\\,dx$是三阶中心矩。记$\rho=(\log f)'$为得分函数，其三阶矩为$J_3=\int f(x)\rho(x)^3\\,dx$。Hermite扰动给出$\mu_3>0$且$J_3<0$，显式余项估计验证了所构造的密度及其端点曲率。该反例不涉及附加对称性假设下的猜想情形。

英文摘要

The Ball-Nayar-Tkocz conjecture asserts that, for independent real random variables $X,Y$ with a common log-concave density, the function $t\mapsto h(\sqrt tX+\sqrt{1-t}Y)$ is concave on $[0,1]$. Gaussian distributions have this property. We construct strongly log-concave counterexamples arbitrarily close to the Gaussian and study their persistence under Gaussian smoothing. Let $X$ have mean zero and variance one, let $G$ be an independent standard Gaussian, and let $q\in[0,1]$ denote the signal variance proportion in the smoothed variable $\sqrt qX+\sqrt{1-q}G$. In the asymmetric case, for every fixed $0<q\le1$, there are counterexamples for which the entropy of the weighted sum of two independent copies of the smoothed variable has strictly positive second derivative near the endpoints. In the symmetric case, for every $4/7<q\le1$, there are counterexamples for which an unequally weighted sum has strictly greater entropy than the equally weighted sum. Both families have smooth, strictly positive densities and are strongly log-concave before and after smoothing. The density ratios $f/φ$ converge uniformly to one, and each derivative of fixed positive order converges uniformly to zero, where $φ$ is the standard Gaussian density. The symmetric counterexamples can also match any prescribed finite number of Gaussian moments exactly. The asymmetric construction uses an endpoint expansion of entropy, whereas the symmetric construction exploits the different decay rates of high-order Hermite perturbations under different weights. Thus, densities violating concavity approach the Gaussian density in the strong sense described above. The question of uniform concavity in the symmetric class for $0<q\le4/7$ remains open.

发表机构

• Fudan University（复旦大学）

机构由 AI 辅助整理，请以论文原文为准。

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37 pages