Quantization of Probability Distributions via Divide-and-Conquer: Convergence and Error Propagation under Distributional Arithmetic Operations

Bilgesu Arif Bilgin, Olof Hallqvist Elias, Michael Selby, Phillip Stanley-Marbell, May 22, 2025

Abstract

This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numerical study that compares pre-existing approximation schemes with a special focus on the stability of the discrete approximations when they undergo arithmetic operations. The main results are a simple upper bound of the approximation error in terms of the Wasserstein-1 distance that is valid for all continuous distributions with finite mean. In many use-cases, the studied method achieve optimal rate of convergence, and numerical experiments show that the algorithm is more stable than pre-existing approximation schemes in the context of arithmetic operations.

Cite as:

Bilgin, Bilgesu Arif, Olof Hallqvist Elias, Michael Selby, and Phillip Stanley-Marbell. "Quantization of Probability Distributions via Divide-and-Conquer: Convergence and Error Propagation under Distributional Arithmetic Operations." arXiv preprint arXiv:2505.15283 (2025).

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