Edward Nelson's book "Radically Elementary Probability Theory" (1987) provides an alternative way for doing probability theory 
based on a nonstandard model of real analysis and finite sample spaces.   Nelson's work does not cover Markov chains. We expand 
his nonstandard theory of stochastic processes with special interest in the limit theory of Markov chains.

The principal results we present relate to Markov chain central limit theorems. We prove a nonstandard version of a CLT for 
strong-mixing stationary sequences (not necessarily Markov chains) and also a CLT for polynomially ergodic chains. A (functional) 
central limit theorem based on the nonstandard invariance principle is also proved.

This is a small step in the development of Markov chains under this nonstandard probability theory. Nelson's radically elementary 
model yields a limit theory for Markov chains with a much simpler mathematical apparatus than the one classically used (e.g. Meyn 
& Tweedie, 1993) and yet useful for advanced applications such as Markov chain Monte Carlo methods in Statistics.