Topics in nonstandard probability theory

This dissertation expands the nonstandard (radically elementary) theory of stochastic processes proposed by Nelson (1987) with special interest in the limit theory of Markov chains. The theory presented here relies on nonstandard analysis instead of measure theory. Nelson's book does not cover Markov chains.;The principal results are found in Sections 4.4 and 5.3 where central limit theorems are proved. We first provide a nonstandard version of a central limit theorem for strongly mixing stationary sequences (not necessarily Markov chains) in Theorem 4.4 and later apply it to a polynomially ergodic Markov chain (Theorem 5.1). A central limit theorem for Markov chains using drift and minorization is proved (Theorem 5.3). A (functional) central limit theorem based on the nonstandard invariance principle is also proved (Theorem 5.4). A brief exposition of the nonstandard probability theory contained in Nelson (1987) and Geyer (2007) is given in the first chapters and a few new results related to stochastic processes, not necessarily Markov chains, are proved.;This is a small step in the development of a theory for Markov chains under Nelson's radically elementary probability theory. An entire radically elementary theory should yield a Markov chain limit theory with a much simpler mathematical apparatus than the classical one (e.g. Meyn & Tweedie, 1993) and yet useful for advanced applications such as Markov chain Monte Carlo methods in Statistics.