稀疏、熵与稀疏数定律
Thinning, Entropy and the Law of Thin Numbers
摘要：
Renyi对离散随机变量的“细化”操作是对连续随机变量的缩放操作的自然离散模拟。在信息理论的背景下，特别是在连接与泊松近似结果相关的信息理论不等式的细化的性质进行了研究。经典的二项式到泊松收敛（有时称为“小数定律”）被视为离散分布卷积的稀疏极限定理的特殊情况。收敛速度提供了这个限制，并建立非渐近界。这种发展在某种程度上与高斯不等式的发展相平行，导致了中心极限定理的信息论版本。特别是，一个“细化马尔可夫链”的介绍，并显示它发挥类似的作用，连接到熵功率不等式的Ornstein-Uhlenbeck过程。
英文摘要：
Renyi's "thinning" operation on a discrete random variable is a natural discrete analog of the scaling operation for continuous random variables. The properties of thinning are investigated in an information-theoretic context, especially in connection with information-theoretic inequalities related to Poisson approximation results. The classical Binomial-to-Poisson convergence (sometimes referred to as the "law of small numbers" is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is provided for this limit, and nonasymptotic bounds are also established. This development parallels, in part, the development of Gaussian inequalities leading to the information-theoretic version of the central limit theorem. In particular, a "thinning Markov chain" is introduced, and it is shown to play a role analogous to that of the Ornstein-Uhlenbeck process in connection to the entropy power inequality.
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