Redundancy of unbounded memory Markov classes with continuity conditions

TL;DR

This paper investigates the limits of universal data compression for binary Markov sources with unbounded memory, focusing on sources that satisfy a continuity condition linking finite-memory estimates to the true infinite-memory distribution.

Contribution

It establishes asymptotically matching bounds on redundancy for Markov sources with continuity conditions, revealing which sources dominate redundancy considerations.

Findings

Derived tight upper and lower bounds on redundancy.

Identified key sources influencing redundancy.

Connected compression performance to source continuity properties.

Abstract

We study the redundancy of universally compressing strings $X_1,\dots, X_n$ generated by a binary Markov source $p$ without any bound on the memory. To better understand the connection between compression and estimation in the Markov regime, we consider a class of Markov sources restricted by a continuity condition. In the absence of an upper bound on memory, the continuity condition implies that $p(X_0|X^{-1}_{-m})$ gets closer to the true probability $p(X_0|X_{-\infty}^{-1})$ as $m$ increases, rather than vary around arbitrarily. For such sources, we prove asymptotically matching upper and lower bounds on the redundancy. In the process, we identify what sources in the class matter the most from a redundancy perspective.