The first digit problem and scale-invariance. (English) Zbl 0853.60002

Volčič, Aljoša

Marcellini, Paolo (ed.) et al., Partial differential equations and applications. Collected papers in honor of Carlo Pucci on the occasion of his 70th birthday. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 177, 329-340 (1996).

The author’s purpose is to give a countably-additive measure-theoretic derivative of Benford’s law \[ \text{Prob[first significant digit(base }10)=k]= \log_{10} [(k+ 1)/ k], \quad k=1, 2, \dots, 9, \] [cf. F. Benford, Proc. Am. Philos. Soc. 78, 551-572 (1938; Zbl 0018.26502); R. A. Raimi, Am. Math. Monthly 83, 521-538 (1976; Zbl 0349.60014)] based on the classical scale-invariance hypothesis. By appropriate choice of measure space, the restriction to Borel subsets of \([1, 10)\), and use of arguments similar to the reviewer’s ones [Proc. Am. Math. Soc. 123, No. 3, 887-895 (1995; Zbl 0813.60002)], it is shown that scale-invariance of the probability measure implies absolute continuity, and Benford’s law then follows from the explicit expression of the distribution function and the resulting density.
For the entire collection see [Zbl 0834.00042].

Reviewer: T.P.Hill (Atlanta)

MSC:

60A10

Probabilistic measure theory

Keywords:

first digit problem; scale-invariance; significant-digit law; Benford’s law

Citations:

Zbl 0018.26502; Zbl 0349.60014; Zbl 0813.60002