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On a Markov Lemma and Typical Sequences for Polish Alphabets

机译：关于马尔可夫引理和波兰语字母的典型序列

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摘要

In this paper, we consider a new definition of typicality based on the weak topology that is applicable to Polish alphabets (which includes ). This notion is a generalization of strong typicality in the sense that it degenerates to strong typicality in the finite alphabet case, and can also be applied to mixed and continuous distributions. Furthermore, it is strong enough to prove a Markov lemma, and thus can be used to directly prove a more general class of results than entropy (or weak) typicality. We provide two example applications of this technique. First, using the Markov Lemma, we directly prove a coding result for Gel’fand–Pinsker channels with an average input constraint for a large class of alphabets and channels without first proving a finite alphabet result and then resorting to delicate quantization arguments. This class of alphabets includes, for example, real and complex inputs subject to a peak amplitude restriction. While this large class does not directly allow for Gaussian distributions with average power constraints, it is shown to be straightforward to recover this case by considering a sequence of truncated Gaussian distributions. As a second example, we consider a problem of coordinated actions (i.e., empirical distributions) for a two node network, where we derive necessary and sufficient conditions for a given desired coordination.
机译：在本文中，我们考虑了适用于波兰语字母（包括）的基于弱拓扑的典型性的新定义。从在有限字母的情况下退化为强典型性的意义上来说，此概念是强典型性的泛化，并且也可以应用于混合分布和连续分布。此外，它足以证明马尔可夫引理，因此可以用来比熵（或弱）典型性直接证明更通用的结果类别。我们提供了此技术的两个示例应用程序。首先，使用马尔可夫引理，我们直接证明了Gel'fand-Pinsker通道的编码结果，该输入结果具有针对一大类字母和通道的平均输入约束，而无需首先证明有限的字母结果，然后诉诸微妙的量化参数。这类字母包括，例如，受峰值幅度限制的实数和复数输入。尽管这一大类不能直接允许具有平均功率约束的高斯分布，但通过考虑一系列截短的高斯分布可以很容易地恢复这种情况。作为第二个例子，我们考虑了两节点网络的协调动作（即经验分布）的问题，其中我们为给定的期望协调导出了必要和充分的条件。

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