A note on special cases derived from Nelson’s martingale central limit theorem

Abstract

This article gives a concise account of the functional central limit theorem (fCLT) found in Nelson’s (nonstandard) radically elementary probability theory (1987, Princeton Univ. Press AMS 117) connecting it to the distributional result (near Gaussianity of the Wiener walk) later established by Benoît. Despite the generality of the fCLT being indicative of the much simpler CLT for $L_2$-IID random variables, such classical CLT has not been stated clearly in the literature of radically elementary probability. Here, a classical CLT is obtained as a corollary of our main result which shows that typical instances used in applications (bounded random variables, $L_2$-IID random sequences and sequences satisfying Lyaponouv’s condition) satisfy the near Lindeberg condition required by the fCLT. The simplicity of the fCLT, as opposed to its conventional counterpart based on highly technical stochastic limit operations, justifies the derivation of the CLT as a special case. The entire work uses Nelson’s radically elementary probability model and it does not require the reader to know the general model, known as internal set theory, upon which the elementary model is based.