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

B. A. Bilgin, K. O. H. Elias, M. Selby, P. Stanley-Marbell, 01 April, 2026

Abstract

This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numeri cal 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 Was serstein-1 distance that is valid for all continuous distributions with finite mean. In many use-cases, the studied method achieves the 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, Karl 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." Methodology and Computing in Applied Probability 28, no. 2 (2026): 32.

Bibtex:

@article{Bilgin_2026, 
      title={Quantization of Probability Distributions via Divide-and-Conquer: Convergence and Error Propagation under Distributional Arithmetic Operations}, 
      volume={28}, 
      ISSN={1573-7713}, 
      url={http://dx.doi.org/10.1007/s11009-026-10261-2}, 
      DOI={10.1007/s11009-026-10261-2}, 
      number={2}, 
      journal={Methodology and Computing in Applied Probability}, 
      publisher={Springer Science and Business Media LLC}, 
      author={Bilgin, Bilgesu Arif and Hallqvist Elias, Karl Olof and Selby, Michael and Stanley-Marbell, Phillip},
      year={2026}, 
      month=apr 
}