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Uniform Distributions on the Natural Numbers

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Abstract

We compare the following three notions of uniformity for a finitely additive probability measure on the set of natural numbers: that it extend limiting relative frequency, that it be shift-invariant, and that it map every residue class mod m to 1/m. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set L of extensions of limiting relative frequency is a proper subset of the set S of shift-invariant measures and that S is a proper subset of the set R of measures which map residue classes uniformly. Moreover, we show that there are subsets G of ℕ for which the range of possible values μ(G) for μ∈L is properly contained in the set of values obtained when μ ranges over S, and that there are subsets G which distinguish S and R analogously.

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Authors and Affiliations

  1. Department of Mathematics, Oberlin College, Oberlin, OH, 44074, USA

    Oliver Schirokauer

  2. Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Joseph B. Kadane

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  1. Oliver Schirokauer
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  2. Joseph B. Kadane
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Correspondence to Joseph B. Kadane.

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Schirokauer, O., Kadane, J.B. Uniform Distributions on the Natural Numbers. J Theor Probab 20, 429–441 (2007). https://doi.org/10.1007/s10959-007-0066-1

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  • DOI: https://doi.org/10.1007/s10959-007-0066-1

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