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Graduate Seminar

Michael Spece
Carnegie Mellon University
Title: Uniformity over the Integers

Abstract: Uniformity is a parsimonious description of randomness. Unfortunately, a probability measure can come nowhere close to approximating uniformity on unbounded sets. This talk takes up that problem in the setting of the integers. By replacing the countable additivity axiom of probability with finite additivity, one can assign to each integer the same probability. Because assigning to each integer the same probability does not uniquely define a uniform distribution, various additional properties have been proposed. In this talk, three such properties will be reviewed, and a new one introduced. It turns out these notions can be ordered so that one implies the next. While stronger notions may be required to attain uniqueness for certain applications, the weaker notions have the virtues of being more parsimonious and more interpretable. Moreover, stronger properties may be decomposed into weaker ones, and the weaker notions may have special number theoretic properties. For instance, the family of uniform distributions introduced in this talk may assign positive probability to the set of all prime numbers.

Date: Thursday, May 9, 2013
Time: 5:30 pm
Location: Wean Hall 8220
Submitted by:  Brian Kell