Four Limits in Probability and Their Roles in Source Coding

Abstract

In information-spectrum methods proposed by Han and Verdú, quantities defined by using the limit superior (or inferior) in probability play crucial roles in many problems in information theory. In this paper, we introduce two nonconventional quantities defined in probabilistic ways. After clarifying basic properties of these quantities, we show that the two quantities have operational meaning in the eps-coding problem of a general source in the ordinary and optimistic senses. The two quantities can be used not only for obtaining variations of the strong converse theorem but also establishing upper and lower bounds on the width of the entropy-spectrum. We also show that the two quantities are expressed in terms of the smooth Rényi entropy of order zero.