Near equivalence on metric spaces and a nonstandard central limit theorem

Abstract

This article proves a nonstandard Central Limit Theorem (CLT) in the
sense of Nelson’s Radically Elementary Probability Theory. The CLT proved
here is obtained by establishing the near equivalence of standardized averages
obtained from L2 IID random variables to the standardized average resulting from a
binomial CLT. A nonstandard model for near equivalence on metric spaces replaces
conventional results of weak convergence. Statements and proofs remain radically
elementary without applying the full Internal Set Theory. A nonstandard notion of
normality is discussed.