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\(n\)-digit benford distributed random variables. (English) Zbl 1286.11114

Summary: The scope of this paper is twofold. First, to emphasize the use of the mod 1 map in exploring the digit distribution of random variables. We show that the well-known base- and scale-invariance of Benford variables are consequences of their associated mod 1 density functions being uniformly distributed. Second, to introduce a new concept of the \(n\)-digit Benford variable. Such a variable is Benford in the first \(n\) digits, but it is not guaranteed to have a logarithmic distribution beyond the \(n\)th digit. We conclude the paper by giving a general construction method for \(n\)-digit Benford variables, and provide a concrete example.

MSC:

11K06 General theory of distribution modulo \(1\)
11K36 Well-distributed sequences and other variations
60E05 Probability distributions: general theory
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