Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

TL;DR

This monograph explores asymptotic estimates in information theory where error probabilities do not vanish, focusing on achievable rates with non-zero error bounds across single- and multi-user scenarios.

Contribution

It provides a unified framework for analyzing information-theoretic problems with non-vanishing error probabilities, including second- and third-order asymptotics.

Findings

Characterization of non-vanishing error probabilities in hypothesis testing

Second- and third-order asymptotic expansions for point-to-point communication

Analysis of network information theory problems with known second-order asymptotics

Abstract

This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.