本福德定律和随机整数分解与全同停止条件

Pub Date : 2024-06-26 DOI:10.1016/j.jnt.2024.05.005
Xinyu Fang , Steven J. Miller , Maxwell Sun , Amanda Verga
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引用次数: 0

摘要

本福德定律是关于数据集中每个数字作为首位数字出现的频率的声明。各种常见的整数序列,如斐波那契数、阶乘和大多数整数的幂,都符合该定律。在本文中,我们证明了由随机积分分解过程(我们将其建模为离散的 "断棒")产生的整数序列,在满足一定的同位停止条件后,会渐近地接近本福分布。我们还证明,我们对定义同余停止条件的同余类数量的要求是本福德行为发生的必要条件,也是一个临界点;偏离这个临界点会导致截然不同的行为。
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Benford's law and random integer decomposition with congruence stopping condition

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete “stick breaking”) subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior.

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