Date of Award

8-2013

Embargo Period

12-20-2013

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Advisor(s)

Jeremy Avigad

Abstract

This dissertation develops connections between algorithmic randomness and computable analysis. In the first part, it is shown that computable randomness can be defined robustly on all computable probability spaces, and that computable randomness is preserved by a.e. computable isomorphisms between spaces. Further applications are also given.

In the second part, a number of almost-everywhere convergence theorems are looked at using computable analysis and algorithmic randomness. These include various martingale convergence theorems and almosteverywhere differentiability theorems. General conditions are given for when the rate of convergence is computable and for when convergence takes place on the Schnorr random points. Examples are provided to show that these almost-everywhere convergence theorems characterize Schnorr randomness.

Included in

Mathematics Commons

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