Breakdown of Edgeworth Expansion in Finite-Blocklength Regime and Exact Absorption via $q$-Deformation
2026-08-24 • Information Theory
Information Theory
AI summaryⓘ
The authors study a problem with a common mathematical tool called the Edgeworth expansion, which sometimes gives impossible negative probabilities when used for communication systems with short message lengths. They introduce a new method using a $q$-deformed framework that changes how probabilities are calculated to avoid these negative values. Their approach carefully adjusts a parameter to fix errors while keeping all probabilities valid. Tests show their method works as well as existing ones but without the downside. This offers a safer way to analyze very reliable communication systems like 6G.
Edgeworth expansionfinite-blocklength regimenegative probabilitiesq-deformationinformation densityskewnessCornish-Fisher boundasymptotic approximationultra-reliable low-latency communication (URLLC)6G communications
Authors
Hiroki Suyari
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.