Benford's law and random integer decomposition with congruence stopping condition (Q6585302)

Benford's law is a statement about the frequency that each digit arises as the leading digit of number in dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, factorials and others.\N\NIn the present article, it is proved that integer sequences resulting from a random integral decomposition process subject to a certain congruence stopping condition satisfy Benford distribution asymptotically. It is shown that a given requirement on the number of congruence classes defining the convergence stopping condition is necessary for Benford behaviour to occur and is a critical point. This article focuses on the Benfordness of sequences of integers resulting from random integer decomposition process with stopping conditions defined by congruence relation.\N\NIn the Introduction, the main concepts of the Benford's law are presented and discussed. The technique of the random integer decomposition is developed. In Theorems 1.1 and 1.2 in the cases of odd and even modulus, the congruence of this decomposition to Benford behaviour is shown.\N\NIn Section 2 the notion of the Benford behaviour is defined. A precise definition for this a sequence of random collection of positive real numbers to converges to strong Benford behaviour is defined.\N\NIn Section 3, the continuous stick breaking process that serves as an approximation to the discrete process is investigated. Theorem 3.1 is the main result in this section. It is shown that the collection of stick lengths after \(N \geq \log R\) level almost surely converges to strong Benford behaviour. The technique of the Mellin transform is recalled and discussed. Also, the Mellin transform condition is developed. In Theorem 3.3, the Mellin transform property is related to the Benford behaviour. In Theorem 3.4 it is given a sufficient condition on \(\mathcal{F}\) for it to satisfy the Mellin transform condition. In this section some families of examples are presented.\N\NIn Subsection 3.3, the proof of Theorem 3.1 under the assumption of independence is developed.\N\NIn Subsection 3.4 how to build the proof of Theorem 3.1 without additional assumptions is explained.\N\NIn Section 4, the setting of discrete stick fragmentation with the convergence stopping condition is considered. Also, the proofs of Propositions 1.4 and 1.5 are presented. In order to carry out the approximation strategy, continuous and discrete processes based on the same sequence of random ratios are defined. Many preliminary results are developed. In Subsection 4.2 the proof of Theorem 1.4 is presented.\N\NIn Section 5, the non-Benfordness of the decomposition process when the stopping condition deviates from the required is proved. It is shown that the conditions, imposed in Theorems 1.2 and 3.1 are necessary. In Subsection 5.1 the continuous case is considered and precise statements of what happens in those cases are proved.\N\NIn Subsection 5.2, the discrete case is considered and a result showing that the final stick lengths are non-Benford. In Theorem 5.6 it is shown that the collection of mantissas of ending stick lengths does not converge to strong Benford behaviour. In Theorem 5.7 it is shown that collection of mantissas of ending stick lengths does not converge to the uniform distribution on \([0,1).\)\N\NIn Section 6, some further direction are considered. The general number of parts and the general stopping condition are discussed. The non-Benfordness for general base is again considered.\N\NIn Appendix A, the proof of Theorem 3.7 is presented.