Empirical Processes, Typical Sequences, and Coordinated Actions in Standard Borel Spaces
Empirical Processes, Typical Sequences, and Coordinated Actions in Standard Borel Spaces
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标准 Borel 空间中的经验过程、典型序列和协调行动
DOI：
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发表时间：
2010
影响因子：
2.5
通讯作者：
M. Raginsky
中科院分区：
文献类型：
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作者：
M. Raginsky
本文提出了一种关于广泛的抽象字母表（所谓的标准 Borel 空间）上的典型序列的新概念，该概念基于通过一类可测量“测试函数”上均匀的经验分布对无记忆源的近似。在有限字母表情况下，我们可以采用所有一致有界函数并恢复强典型性（或总变异距离下的典型性）的通常概念。然而，对于通用字母表来说，这个函数类太大了，必须受到限制。考虑到这一点，我们定义了任何 Glivenko-Cantelli 函数类（即，承认大数统一定律的函数类）的典型性，并通过对几种源编码场景中可实现速率的基本限制进行简单推导来证明其威力，其中相关操作标准涉及再现通用字母表固定无记忆源相对于合适函数类的经验平均值。
This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable “test functions.” In the finite-alphabet case, we can take all uniformly bounded functions and recover the usual notion of strong typicality (or typicality under the total variation distance). For a general alphabet, however, this function class turns out to be too large, and must be restricted. With this in mind, we define typicality with respect to any Glivenko-Cantelli function class (i.e., a function class that admits a Uniform Law of Large Numbers) and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios, in which the relevant operational criteria pertain to reproducing empirical averages of a general-alphabet stationary memoryless source with respect to a suitable function class.