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.
Highlights
Nelson’s [11] book “Radically Elementary Probability Theory” (REPT) provides a nonstandard probability model based on an axiomatic subsystem of Internal Set Theory (Nelson [10]) known as minimal IST (Herzberg [8])
Recent advances in IST-based probability theory include diffusions and interacting particle systems (Weisshaupt [15, 16]), Markov chains (Andrade [2]), discrete functions on infinitesimal grids with an application to probability, stochastic calculus including Ito’s stochastic integration and Levy processes [8] and several applications covered in the book edited by Diener and Diener [6]
The main contribution of this paper is to prove a classical Central Limit Theorem (CLT) (Theorem 3.2) directly without reference to the functional central limit theorem (fCLT)
Summary
Nelson’s [11] book “Radically Elementary Probability Theory” (REPT) provides a nonstandard probability model based on an axiomatic subsystem of Internal Set Theory (Nelson [10]) known as minimal IST (Herzberg [8]). REPT has a functional central limit theorem (fCLT) [11, Theorem 18.1] where the objects of interest are martingales and the Wiener walk (nonstandard Brownian motion) This fCLT has been specialized by Andrade [3] to a classical CLT for independent and identically distributed (IID) nonstandard L2 random variables (rvs). The main contribution of this paper is to prove a classical CLT (Theorem 3.2) directly without reference to the fCLT An important goal of REPT is to be within reach by anyone familiar with basic (discrete) probability and the minimal notions about the rigorous use of infinitesimals (Nelson’s “tiny bit” analysis [11, Preface]) without any heavy apparatus from nonstandard analysis, whether from IST [10] or from Robinson’s nonstandard model [13]
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