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

2026-08-24

AI Summary

本文解决了Edgeworth展开在有限码长下产生负概率的问题，通过q-变形方法调整信息密度空间，消除了三阶偏度并保持了非负性。

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.

Problem

• Edgeworth expansion
• finite-blocklength regime
• negative probabilities
• deep-tail region

Innovation

• q-deformation
• finite-blocklength regime
• nonnegativity
• third-order skewness
• ultra-reliable communications