How Many Digits are Needed?

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-02-08 DOI:10.1007/s11009-024-10073-2

Ira W. Herbst, Jesper Møller, Anne Marie Svane

引用次数: 0

Abstract

Let \(X_1,X_2,...\) be the digits in the base-q expansion of a random variable X defined on [0, 1) where \(q\ge 2\) is an integer. For \(n=1,2,...\), we study the probability distribution \(P_n\) of the (scaled) remainder \(T^n(X)=\sum _{k=n+1}^\infty X_k q^{n-k}\): If X has an absolutely continuous CDF then \(P_n\) converges in the total variation metric to the Lebesgue measure \(\mu \) 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 \(\mu \) 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.

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需要多少位数？

让\(X_1,X_2,...\)是定义在[0, 1]上的随机变量X的基q展开中的数字，其中\(q\ge 2\) 是整数。对于 n=1,2,...\)，我们研究（缩放）余数 \(T^n(X)=sum _{k=n+1}^\infty X_k q^{n-k}\) 的概率分布 \(P_n\)：如果 X 有一个绝对连续的 CDF，那么 \(P_n\) 在总变化度量中收敛于单位区间上的 Lebesgue 度量 \(\mu\) 。在弱平稳条件下，我们首先建立了X和一个非负整数值随机变量N之间的耦合，这样\(T^N(X)\)就跟随\(\mu \)并且与\((X_1,...,X_N)\)无关，其次\(P_n\)和它的PDF\(f_n\)呈指数级快速收敛。我们将讨论需要多少位数，并举例说明我们的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.