The Multivariate Covering Lemma and its Converse

TL;DR

This paper proves a version of the multivariate covering lemma for weakly typical sets, enabling achievability proofs for continuous channels like Gaussian without quantization, and discusses its implications in information theory.

Contribution

It provides a proof of the multivariate covering lemma for weakly typical sets applicable to continuous channels, expanding its utility in information theory.

Findings

Proof of the covering lemma for weakly typical sets.

Enables achievability proofs for Gaussian channels without quantization.

Broad applicability in multi-user information theory scenarios.

Abstract

The multivariate covering lemma states that given a collection of $k$ codebooks, each of sufficiently large cardinality and independently generated according to one of the marginals of a joint distribution, one can always choose one codeword from each codebook such that the resulting $k$-tuple of codewords is jointly typical with respect to the joint distribution. We give a proof of this lemma for weakly typical sets. This allows achievability proofs that rely on the covering lemma to go through for continuous channels (e.g., Gaussian) without the need for quantization. The covering lemma and its converse are widely used in information theory, including in rate-distortion theory and in achievability results for multi-user channels.