814

# Finitely Additive Uniform Distributions on the Natural Numbers: Shift-Invariance

Oliver Schirokauer and Joseph B. Kadane

Updated: February 2006

### 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 to . We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set of extensions of limiting relative frequency is a proper subset of the set of shift-invariant measures and that is a proper subset of the set of measures which map residue classes uniformly. Moreover, we show that there are subsets of for which the range of possible values for is properly contained in the set of values obtained when ranges over , and that there are subsets which distinguish and analogously.

Keywords:imit points, limiting relative frequency, non-conglomerability, probability charge, residue class, shift-invariance

Heidi Sestrich 2005-02-02
Here is the full PDF text for this technical report. It is 162870 bytes long.