Shepp-Olkin熵谱猜想的一个证明
A proof of the Shepp-Olkin entropy concavity conjecture
摘要：
我们证明了Shepp-Olkin猜想，它指出独立Bernoulli随机变量之和的熵在单个随机变量的参数中是凹的。我们的证明是以前提出的同一作者，解决了单调的情况下（其中所有的参数同时增加）的猜想参数的改进。事实上，我们表明，单调的情况下是最坏的情况下，使用仔细分析的概率质量函数的导数的性质。我们提出了一个推广的Shepp和Olkin的原始猜想，考虑Renyi和Tsallis熵。
英文摘要：
We prove the Shepp--Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof is a refinement of an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin's original conjecture, to consider Renyi and Tsallis entropies.
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