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Some remarks on the first digit problem. (English) Zbl 0912.60001
The problem of the first significant digit can be described as follows: in a very large sample of positive real numbers, randomly chosen, if the frequency of the first non-zero digit is tested, it appears that the occurrence of the 9 digits is not the same for all. The experiments so far performed agree with the so-called “Benford law”: if $$f(j)$$ denotes the frequency of the digit $$j$$, then $$f(j)= \text{Log} (j+1) -\text{Log} (j)$$, with $$\text{Log} =\log_{10}$$. The typical way to approach the first-digit problem is to denote by $$Y$$ the “random variable” resulting from the experiments, i.e. $$Y$$ is any positive real number, completely randomly chosen, and to describe the distribution of $$Y$$, in order to deduce under suitable conditions that the first significant digit of $$Y$$ obeys the Benford law. The main purpose of the present paper is to describe the “distribution” of $$Y$$, first introducing some models and then deducing the Benford-compatibility from some kinds of invariance. Section 1 of the paper introduces the classes of functions that satisfy suitable limit conditions and relates these properties to weak*-convergence of an appropriate net of measurable functions. Section 2 examines the convergence to Benford-compatible distributions for suitable sequences $$(Y_n)$$ associated to $$Y$$. Section 3 defines Benford-compatibility and suggests a model for $$Y$$ as a limit of suitable sequences $$(Y_n)$$. Finally, two different types of invariance are defined, power-invariance and $$r$$-scale-invariance, and it is proved that power-invariance is equivalent to Benford-compatibility, while $$r$$-scale-invariance is equivalent to a weaker form of Benford-compatibility.
Reviewer: N.Curteanu (Iaşi)

##### MSC:
 60B10 Convergence of probability measures 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)