Minimax redundancy for Markov chains with large state space

TL;DR

This paper analyzes the rate at which universal coding approaches the Shannon limit for large-state Markov sources, revealing a phase transition in sample complexity related to the alphabet size and mixing time.

Contribution

It establishes a precise phase transition point for the sample size needed to achieve vanishing redundancy in Markov sources with large state spaces.

Findings

Redundancy vanishes at a rate depending on alphabet size and mixing time.

Identifies a phase transition at sample size proportional to the square of the state space.

Provides bounds on the sample complexity for near-optimal compression.

Abstract

For any Markov source, there exist universal codes whose normalized codelength approaches the Shannon limit asymptotically as the number of samples goes to infinity. This paper investigates how fast the gap between the normalized codelength of the "best" universal compressor and the Shannon limit (i.e. the compression redundancy) vanishes non-asymptotically in terms of the alphabet size and mixing time of the Markov source. We show that, for Markov sources whose relaxation time is at least $1 + \frac{(2+c)}{\sqrt{k}}$, where $k$ is the state space size (and $c>0$ is a constant), the phase transition for the number of samples required to achieve vanishing compression redundancy is precisely $\Theta(k^2)$.