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Breakdown of Edgeworth Expansion in Finite-Blocklength Regime and Exact Absorption via q-Deformation

  • Hiroki Suyari
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Abstract

This paper addresses the structural breakdown of the Edgeworth expansion in the finite-blocklength (FBL) regime, where conventional asymptotic approximations yield unphysical negative probabilities in the deep-tail region. We propose a q-deformed framework that resolves this inconsistency by replacing additive polynomial perturbations with a geometric deformation of the information density space. Motivated by the linearization of nonlinear dynamics, we prove that dynamically scaling the q-logarithmic parameter exactly absorbs the third-order skewness while preserving global nonnegativity. We establish a universal asymptotic matching, demonstrating that the framework encapsulates higher-order asymptotic scales. Numerical results confirm that the proposed method matches the state-of-the-art precision of the Cornish-Fisher bound without the risk of negative probabilities. The framework offers a robust and computationally stable foundation for evaluating operational limits in ultra-reliable communications such as 6G and URLLC.

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Cite

BibTeXRIS
BibTeX
@misc{suyari2026breakdown,
  title = {{Breakdown of Edgeworth Expansion in Finite-Blocklength Regime and Exact Absorption via $q$-Deformation}},
  author = {Hiroki Suyari},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.23303},
  eprint = {2608.23303},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23303},
}
RIS
TY  - UNPB
TI  - Breakdown of Edgeworth Expansion in Finite-Blocklength Regime and Exact Absorption via q-Deformation
AU  - Hiroki Suyari
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.23303
UR  - https://arxiv.org/abs/2608.23303
AB  - This paper addresses the structural breakdown of the Edgeworth expansion in the finite-blocklength (FBL) regime, where conventional asymptotic approximations yield unphysical negative probabilities in the deep-tail region. We propose a q-deformed framework that resolves this inconsistency by replacing additive polynomial perturbations with a geometric deformation of the information density space. Motivated by the linearization of nonlinear dynamics, we prove that dynamically scaling the q-logarithmic parameter exactly absorbs the third-order skewness while preserving global nonnegativity. We establish a universal asymptotic matching, demonstrating that the framework encapsulates higher-order asymptotic scales. Numerical results confirm that the proposed method matches the state-of-the-art precision of the Cornish-Fisher bound without the risk of negative probabilities. The framework offers a robust and computationally stable foundation for evaluating operational limits in ultra-reliable communications such as 6G and URLLC.
ER  -