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Moment-optimal finitary isomorphism for i.i.d. processes of equal entropy

Uri Gabor

TL;DR

The paper resolves the open question of moment bounds for finitary isomorphisms between i.i.d. processes of equal entropy by proving that moment tails of the coding radius can be made finite for all moments with exponent $<\tfrac{1}{2}$ in the i.i.d. (and more generally aperiodic Markov) setting. It introduces marker processes and skeleton-based partitioning to build finitary encodings with optimal tail behavior, then develops a robust framework of good Systems of Fillers (SOF) and societies to realize explicit, measurable isomorphisms via Keane–Smorodinsky coding. The authors provide a complete construction that yields explicit tail bounds $P[R>n]=\tilde{O}(n^{-1/2})$ for both the coding radius and its inverse, establishing entropy as a full invariant for finitary isomorphism with finite sub-half moments. The general result extends to arbitrary finite-alphabet aperiodic Markov shifts of equal entropy by chaining a sequence of bi-marked-process steps, preserving the tail decay through careful control of filler equivalence classes. Collectively, the work delivers a moment-optimal, explicit, finitary isomorphism framework with broad implications for entropy-based classification in ergodic theory and related coding problems.

Abstract

The finitary isomorphism theorem, due to Keane and Smorodinsky, raised the natural question of how "finite" the isomorphism can be, in terms of moments of the coding radius. More precisely, for which values does there exist an isomorphism between any two i.i.d. processes of equal entropy, with coding radii exhibiting finite t-moments? [3, 4]. Parry [13] and Krieger [10] showed that those finite moments must be lesser than 1 in general, and Harvey and Peres [5] showed that they must be lesser than 1/2 in general. However, the question for the range between 0 and 1/2 remained open, and in fact no general construction of an isomorphism was shown to exhibit any non trivial finite moments. In the present work we settle this problem, showing that between any two aperiodic Markov processes (and i.i.d. processes in particular) of the same entropy, there exists an isomorphism f with coding radii exhibiting finite t-moments for all t in (0,1/2). The isomorphism is constructed explicitly, and the tails of the radii are shown to be optimal up to a poly-logarithmic factor.

Moment-optimal finitary isomorphism for i.i.d. processes of equal entropy

TL;DR

The paper resolves the open question of moment bounds for finitary isomorphisms between i.i.d. processes of equal entropy by proving that moment tails of the coding radius can be made finite for all moments with exponent in the i.i.d. (and more generally aperiodic Markov) setting. It introduces marker processes and skeleton-based partitioning to build finitary encodings with optimal tail behavior, then develops a robust framework of good Systems of Fillers (SOF) and societies to realize explicit, measurable isomorphisms via Keane–Smorodinsky coding. The authors provide a complete construction that yields explicit tail bounds for both the coding radius and its inverse, establishing entropy as a full invariant for finitary isomorphism with finite sub-half moments. The general result extends to arbitrary finite-alphabet aperiodic Markov shifts of equal entropy by chaining a sequence of bi-marked-process steps, preserving the tail decay through careful control of filler equivalence classes. Collectively, the work delivers a moment-optimal, explicit, finitary isomorphism framework with broad implications for entropy-based classification in ergodic theory and related coding problems.

Abstract

The finitary isomorphism theorem, due to Keane and Smorodinsky, raised the natural question of how "finite" the isomorphism can be, in terms of moments of the coding radius. More precisely, for which values does there exist an isomorphism between any two i.i.d. processes of equal entropy, with coding radii exhibiting finite t-moments? [3, 4]. Parry [13] and Krieger [10] showed that those finite moments must be lesser than 1 in general, and Harvey and Peres [5] showed that they must be lesser than 1/2 in general. However, the question for the range between 0 and 1/2 remained open, and in fact no general construction of an isomorphism was shown to exhibit any non trivial finite moments. In the present work we settle this problem, showing that between any two aperiodic Markov processes (and i.i.d. processes in particular) of the same entropy, there exists an isomorphism f with coding radii exhibiting finite t-moments for all t in (0,1/2). The isomorphism is constructed explicitly, and the tails of the radii are shown to be optimal up to a poly-logarithmic factor.

Paper Structure

This paper contains 26 sections, 34 theorems, 343 equations.

Table of Contents

  1. Marker processes
  2. Definitions and a lemma
  3. Construction of the marker process $\xi$
  4. Proof of lemma \ref{['lem:Marker=000020process']}
  5. Two processes with a common state
  6. Fillers and systems of fillers (SOF)
  7. Azuma-Hoeffding inequality for Markov chains and renewal processes
  8. The isomorphism construction
  9. Societies
  10. Keane-Smorodinsky coding
  11. Construction of Good SOF
  12. Proof outline of Proposition \ref{['prop:Existence=000020of=000020good=000020filler=000020systems']}.
  13. Construction of a good SOF $\mathcal{A}$
  14. Proof of Proposition \ref{['prop:Existence=000020of=000020good=000020filler=000020systems']}:
  15. Proof of Theorem \ref{['thm:common=000020entry=000020version']}:
  16. ...and 11 more sections

Key Result

Theorem 1

For any two finite alphabet aperiodic Markov shifts $X\sim(A^{\mathbb{Z}},\mu),Y\sim(B^{\mathbb{Z}},\nu)$ of the same entropy, there exists a finitary isomorphism $\phi:A^{\mathbb{Z}}\rightarrow B^{\mathbb{Z}}$ with coding radii $R_{\phi},R_{\phi^{-1}}$ satisfying: Consequently, both coding radii achieve finite $\theta$-moment for all $\theta<\frac{1}{2}$.

Theorems & Definitions (65)

  • Theorem 1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • Corollary 1.6
  • Definition 1.7
  • Proposition 1.8
  • proof
  • ...and 55 more