Doctor of Philosophy (PhD)
This dissertation broadly deals with two areas of probability theory and investigates how methods from nonstandard analysis may provide new perspectives in these topics. In particular, we use nonstandard analysis to prove new results in the topics of limiting spherical integrals and of exchangeability.
In the former area, our methods allow us to represent finite dimensional Gaussian measures in terms of marginals of measures on hyperfinite-dimensional spheres in a certain strong sense, thus generalizing some previously known results on Gaussian Radon transforms as limits of spherical integrals. This first area has roots in the kinetic theory of gases, which is also described.
In the latter area, we prove a new generalization of de Finetti's theorem for exchangeable random variables, a theorem important for the foundations of Bayesian statistics. In particular, we extend the de Finetti--Hewitt--Savage theorem to certain general sequences of exchangeable random variables taking values in any Hausdorff space. Under mild distributional conditions, our work expresses a sequence of exchangeable random variables taking values in any Hausdorff space as a mixture of sequences of iid random variables. Prior to this work, this result was known for random variables taking values in a Polish space. Hence, the current work has removed the need to have any assumptions on the state space, and shown that it is the underlying distribution of the random variables that is important. We prove several preparatory results in nonstandard and topological measure theory along the way, a highlight being a new generalization of Prokhorov's theorem.
Alam, Irfan, "Applications of Nonstandard Analysis in Probability and Measure Theory" (2021). LSU Doctoral Dissertations. 5554.