Uniform Distributions on the Natural Numbers

Oliver Schirokauer and Joseph B. Kadane ()
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Oliver Schirokauer: Oberlin College
Joseph B. Kadane: Carnegie Mellon University

Journal of Theoretical Probability, 2007, vol. 20, issue 3, 429-441

Abstract: 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.

Keywords: Limit points; Limiting relative frequency; Non-conglomerability; Probability charge; Residue class; Shift-invariance (search for similar items in EconPapers)
Date: 2007
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10959-007-0066-1

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