How Many Digits are Needed?

Let X 1 , X 2 , . . . 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 T n ( X ) = ∑ k = n + 1 ∞ X k q n - k : If X has an absolutely continuous CDF t...

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Bibliographic Details
Published in:Methodology and computing in applied probability 2024-03, Vol.26 (1), p.5, Article 5
Main Authors: Herbst, Ira W., Møller, Jesper, Svane, Anne Marie
Format: Article
Language:English
Subjects:
Business and Management
Codes
Convergence
Digits
Economics
Electrical Engineering
Integers
Life Sciences
Mathematics and Statistics
Probability distribution
Random variables
Smoothness
Statistics
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Summary:Let X 1 , X 2 , . . . 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 T n ( 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 ( X 1 , . . . , 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.
ISSN:1387-5841
1573-7713
DOI:10.1007/s11009-024-10073-2