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Uniform Distributions on the Natural Numbers

Author

Listed:
  • Oliver Schirokauer

    (Oberlin College)

  • Joseph B. Kadane

    (Carnegie Mellon University)

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.

Suggested Citation

  • Oliver Schirokauer & Joseph B. Kadane, 2007. "Uniform Distributions on the Natural Numbers," Journal of Theoretical Probability, Springer, vol. 20(3), pages 429-441, September.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:3:d:10.1007_s10959-007-0066-1
    DOI: 10.1007/s10959-007-0066-1
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    Citations

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    Cited by:

    1. Ryoichi Kunisada, 2022. "On the Additive Property of Finitely Additive Measures," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1782-1794, September.
    2. Efe A. Ok & Andrei Savochkin, 2022. "Believing in forecasts, uncertainty, and rational expectations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 74(3), pages 947-971, October.
    3. Timber Kerkvliet & Ronald Meester, 2016. "Uniquely Determined Uniform Probability on the Natural Numbers," Journal of Theoretical Probability, Springer, vol. 29(3), pages 797-825, September.

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