独立实验熵的再研究

摘要：

弱大数定律意味着，在对源的温和假设下，每个产生的符号的Renyi熵（概率）收敛于香农熵率。本文量化的速度，这种收敛的源独立（但不是iid）的输出，推广和改进的结果Holenstein和Renner（IEEE Trans. Inform. Theory，2011）. (a)我们用\emph{slowest convergence}（对于给定的熵）来表征源：它们的输出是均匀分布和单位质量的混合。(b)基于上述特征，我们在\emph{high-entropy}体系中建立了更快的收敛。我们将讨论如何使用这些改进的界限，以更好地量化安全的随机数发生器的输出。反过来，“最坏”分布的特征可以用来导出Renyi和Shannon熵之间的尖锐的“极值”不等式。主要的技术是\emph{non-convex programming}，用于表征在一定熵下可能大的指数矩的分布。

英文摘要：

The weak law of large numbers implies that, under mild assumptions on the source, the Renyi entropy per produced symbol converges (in probability) towards the Shannon entropy rate. This paper quantifies the speed of this convergence for sources with independent (but not iid) outputs, generalizing and improving the result of Holenstein and Renner (IEEE Trans. Inform. Theory, 2011). (a) we characterize sources with \emph{slowest convergence} (for given entropy): their outputs are mixtures of a uniform distribution and a unit mass. (b) based on the above characterization, we establish faster convergences in \emph{high-entropy} regimes. We discuss how these improved bounds may be used to better quantify security of outputs of random number generators. In turn, the characterization of "worst" distributions can be used to derive sharp "extremal" inequalities between Renyi and Shannon entropy. The main technique is \emph{non-convex programming}, used to characterize distributions of possibly large exponential moments under certain entropy.

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