q-期望值和Renyi熵的鲁棒性
On the robustness of q-expectation values and Renyi entropy
摘要：
本文研究了概率分布的泛函（如Rényi熵和非加性S_q熵）以及q-期望值在分布的小变化下的稳健性。我们集中在三个重要类型的分布函数，即（i）连续有界（ii）离散与有限数量的状态，和（iii）离散与无限数量的状态。鲁棒性的物理概念与泛函的数学上更强的稳定性和Lesche稳定性条件进行了对比。我们证明了在连续分布的情形下，一旦排除无界分布和导致负熵的分布，Renyi熵和非加性S_q熵以及q期望值都是鲁棒的.对于离散有限情形，Renyi熵和非可加S_q熵以及q期望值是鲁棒的。对于无限离散的情况下，Renyi熵和q-期望分别违反Lesche稳定性和稳定性，我们表明，一个仍然可以状态的条件，保证物理鲁棒性。
英文摘要：
We study the robustness of functionals of probability distributions such as the Rényi and nonadditive S_q entropies, as well as the q-expectation values under small variations of the distributions. We focus on three important types of distribution functions, namely (i) continuous bounded (ii) discrete with finite number of states, and (iii) discrete with infinite number of states. The physical concept of robustness is contrasted with the mathematically stronger condition of stability and Lesche-stability for functionals. We explicitly demonstrate that, in the case of continuous distributions, once unbounded distributions and those leading to negative entropy are excluded, both Renyi and nonadditive S_q entropies as well as the q-expectation values are robust. For the discrete finite case, the Renyi and nonadditive S_q entropies and the q-expectation values are robust. For the infinite discrete case, where both Renyi entropy and q-expectations are known to violate Lesche-stability and stability respectively, we show that one can nevertheless state conditions which guarantee physical robustness.
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