How Many Digits are Needed?

Ira W. Herbst (), Jesper Møller () and Anne Marie Svane ()
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Ira W. Herbst: University of Virginia
Jesper Møller: Aalborg University
Anne Marie Svane: Aalborg University

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 1, 1-13

Abstract: Abstract Let $$X_1,X_2,...$$ X 1 , X 2 , . . . be the digits in the base-q expansion of a random variable X defined on [0, 1) where $$q\ge 2$$ q ≥ 2 is an integer. For $$n=1,2,...$$ n = 1 , 2 , . . . , we study the probability distribution $$P_n$$ P n of the (scaled) remainder $$T^n(X)=\sum _{k=n+1}^\infty X_k q^{n-k}$$ T n ( X ) = ∑ k = n + 1 ∞ X k q n - k : If X has an absolutely continuous CDF then $$P_n$$ P n converges in the total variation metric to the Lebesgue measure $$\mu $$ 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 $$T^N(X)$$ T N ( X ) follows $$\mu $$ μ and is independent of $$(X_1,...,X_N)$$ ( X 1 , . . . , X N ) , and second exponentially fast convergence of $$P_n$$ P n and its PDF $$f_n$$ f n . We discuss how many digits are needed and show examples of our results.

Keywords: Asymptotic distribution; Coupling; Exponential convergence rate; Extended Newcomb-Benford law; Remainder of a digit expansion; Total variation distance; 60F25; 62E17; 37A50 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-024-10073-2

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