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Unified Algebraic Absorption of Finite-Blocklength Penalties via Generalized Logarithmic Mapping

Hiroki Suyari

Abstract

In finite-blocklength information theory, evaluating the fundamental limits of channel coding typically relies on normal approximations and Edgeworth expansions, which introduce additive polynomial corrections for skewness and higher-order moments. This paper proposes an alternative approach: rather than appending external error terms to a Gaussian baseline, we absorb these finite-length penalties using a generalized $q$-algebraic framework. By introducing a dynamic scaling law $1-q_n = αn^{-1}$ for the tuning parameter, we prove that the $q$-generalized information density corresponds to macroscopic higher-order fluctuations. Specifically, by setting this scaling constant to $α= T/(3V^2)$ (where $V$ is the varentropy and $T$ is the third central moment), our framework recovers the third-order coding limit, absorbing the $O(1)$ non-Gaussian penalty without relying on Hermite polynomials. Furthermore, we demonstrate that the $k$-th degree term of our algebraic expansion matches the $O(n^{1-k/2})$ asymptotic order of the $(k+1)$-th moment Edgeworth correction. This approach unifies classical probabilistic approximations within a single algebraic structure, establishing a mathematical connection between finite-blocklength analysis and generalized logarithmic mappings.

Unified Algebraic Absorption of Finite-Blocklength Penalties via Generalized Logarithmic Mapping

Abstract

In finite-blocklength information theory, evaluating the fundamental limits of channel coding typically relies on normal approximations and Edgeworth expansions, which introduce additive polynomial corrections for skewness and higher-order moments. This paper proposes an alternative approach: rather than appending external error terms to a Gaussian baseline, we absorb these finite-length penalties using a generalized -algebraic framework. By introducing a dynamic scaling law for the tuning parameter, we prove that the -generalized information density corresponds to macroscopic higher-order fluctuations. Specifically, by setting this scaling constant to (where is the varentropy and is the third central moment), our framework recovers the third-order coding limit, absorbing the non-Gaussian penalty without relying on Hermite polynomials. Furthermore, we demonstrate that the -th degree term of our algebraic expansion matches the asymptotic order of the -th moment Edgeworth correction. This approach unifies classical probabilistic approximations within a single algebraic structure, establishing a mathematical connection between finite-blocklength analysis and generalized logarithmic mappings.
Paper Structure (20 sections, 3 theorems, 25 equations, 1 figure)

This paper contains 20 sections, 3 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: Finite-Blocklength Information Theory
  3. Information Density and Varentropy
  4. Second-Order Asymptotics
  5. Limitations of the Normal Approximation
  6. The $q$-Algebraic Framework for Finite-Blocklength
  7. Blocklength Extension and $q$-Information Density
  8. Dynamic Scaling of the $q$-Parameter
  9. Mathematical Proof of the Second-Order Coding Rate
  10. The Renormalization of Higher-Order Moments
  11. Information Spectrum under $q$-Algebra
  12. Matching the Edgeworth Expansion
  13. Numerical Validation for the Second-Order Asymptotics
  14. Simulation Setup and the Canonical Problem
  15. Discussion: The Second-Order Discrepancy
  16. ...and 5 more sections

Key Result

Proposition 3

To generate the finite-length $O(1)$ skewness correction directly from the algebraic structure of the generalized information density, the tuning parameter $(1-q_n)$ must scale as:

Figures (1)

  • Figure 1: Unified comparison of the exact finite-blocklength limit, normal approximation, Edgeworth expansion, and the proposed $q$-algebraic bound. The inset highlights the short-blocklength regime ($n \in [20, 50]$), demonstrating the algebraic absorption of the skewness penalty.

Theorems & Definitions (7)

  • Remark 1: Notation on Entropy and Information Density
  • Definition 2: Centralized $q$-Generalized Information Density
  • Proposition 3: Dynamic Scaling Law for Skewness Absorption
  • Remark 4: Absorption of Kurtosis
  • Definition 5: $q$-Generalized Information Spectrum Limit
  • Theorem 6: Algebraic Absorption of Skewness
  • Theorem 7: Algebraic Resonance of Asymptotic Orders