How many digits are needed?

2023/07/13 by Ira Herbst, Herbst, Ira W., Jesper Møller +3 · 1 citation
Mathematics · #37A50 (Secondary) #60F25 (Primary) 62E17 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2307.06685

openalex publication_date 2023/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X₁,X₂,... be the digits in the base-q expansion of a random variable X defined on [0,1) where q≥2 is an integer. For n=1,2,..., we study the probability distribution P_n of the (scaled) remainder Tⁿ(X)=∑_k=n+1^∞ X_k q^n-k: If X has an absolutely continuous CDF then P_n converges in the total variation metric to the Lebesgue measure μ on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that T^N(X) follows μ and is independent of (X₁,...,X_N), and second exponentially fast convergence of P_n and its PDF f_n. We discuss how many digits are needed and show examples of our results. The convergence results are extended to the case of a multivariate random variable defined on a unit cube.

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• Sufficient digits and density estimation: A Bayesian nonparametric approach using generalized finite Pólya trees

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