Thinning and the Law of Small Numbers
Thinning and the Law of Small Numbers
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细化和小数定律
DOI：
10.1109/isit.2007.4557433
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
Ioannis Kontoyiannis
中科院分区：
文献类型：
--
作者：
P. Harremoës;O. Johnson;Ioannis Kontoyiannis
对离散随机变量的“细化”操作是缩放连续变量的自然离散模拟，即将其乘以一个常量。我们在泊松近似的信息论不等式的背景下研究了稀疏的作用和性质。经典的二项式到泊松收敛，通常被称为“小数定律”，被视为离散分布卷积的稀疏极限定理的特例。对于这一限制，还提供了收敛速度。建立了通道博弈的纳什均衡，其中泊松噪声和泊松输入是最优策略。我们的发展在一定程度上类似于高斯不等式的发展，导致了中心极限定理的信息论版本。
The "thinning" operation on a discrete random variable is the natural discrete analog of scaling a continuous variable, i.e., multiplying it by a constant. We examine the role and properties of thinning in the context of information-theoretic inequalities for Poisson approximation. The classical Binomial-to-Poisson convergence, often 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 also provided for this limit. A Nash equilibrium is established for a channel game, where Poisson noise and a Poisson input are optimal strategies. Our development partly parallels the development of Gaussian inequalities leading to the information- theoretic version of the central limit theorem.