Approximate Uniformity in Finite Convolution Models

2026-09-29

AI Summary

This study addresses the problem of approximating a uniform distribution using the sum of two independent random variables with fixed supports. Methodologically, the approximation performance is systematically analyzed under multiple metrics, including Manhattan distance, Euclidean distance, and Kullback-Leibler divergence, with explicit results derived for specific convolution models. Furthermore, a novel reconstruction method is proposed to address the conjecture on the coding capacity of additive noise channels. The main contributions include obtaining closed-form conclusions for two classes of convolution models and successfully verifying the aforementioned capacity conjecture through the proposed reconstruction approach. These findings provide new theoretical tools for related problems in probability theory and channel coding.

Abstract

The problem of the uniform law being a sum of two independent distributions has been well studied. Here, we study the approximation of the uniform law to a sum of two distributions with fixed support, under the following discrepancies: the Manhattan distance, the Euclidean distance, the forward Kullback--Leibler divergence and the Wasserstein-one distance based on the line metric. The problem of membership of the uniform law in this model has been well studied. Explicit results are obtained in two models, one where one of the distributions is a Bernoulli and the other when both distributions have the same support, including one conjecture. Finally, we show an application of our reconstruction method to a recent conjecture about the coding capacity of an additive noise channel.

Problem

• Approximate Uniformity
• Finite Convolution Models
• Uniform Law
• Discrepancy Measures
• Additive Noise Channel

Innovation

• Finite Convolution Models
• Approximate Uniformity
• Kullback-Leibler Divergence
• Wasserstein Distance
• Coding Capacity