Second-Order Resolvability, Intrinsic Randomness, and Fixed-Length Source Coding for Mixed Sources: Information Spectrum Approach

TL;DR

This paper analyzes second-order asymptotics for mixed sources in information spectrum problems, providing explicit computation methods for nonergodic sources like mixtures of i.i.d. sources, extending previous results.

Contribution

It introduces explicit second-order achievable rate formulas for mixed sources, including nonergodic cases, using two-peak asymptotic normality, extending prior asymptotic normality results.

Findings

Explicit second-order rates for mixed i.i.d. sources

Extension to countably infinite and continuous mixtures

Application of two-peak asymptotic normality

Abstract

The second-order achievable asymptotics in typical random number generation problems such as resolvability, intrinsic randomness, fixed-length source coding are considered. In these problems, several researchers have derived the first-order and the second-order achievability rates for general sources using the information spectrum methods. Although these formulas are general, their computation are quite hard. Hence, an attempt to address explicit computation problems of achievable rates is meaningful. In particular, for i.i.d. sources, the second-order achievable rates have earlier been determined simply by using the asymptotic normality. In this paper, we consider mixed sources of two i.i.d. sources. The mixed source is a typical case of nonergodic sources and whose self-information does not have the asymptotic normality. Nonetheless, we can explicitly compute the second-order achievable rates for these sources on the basis of two-peak asymptotic normality. In addition, extensions of our results to more general mixed sources, such as a mixture of countably infinite i.i.d. sources or Markovian sources, and a continuous mixture of i.i.d. sources, are considered.