On a Markov Lemma and Typical Sequences for Polish Alphabets

Abstract

In this paper, we consider a new definition of typicality based on the weak* topology that is applicable to Polish alphabets (which includes $ \mathbb {R}^{n}$ ). 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.