Convergence determining sets in the central limit theorem (Q1076407)

Let \(X_ 1,X_ 2,..\). be independent and identically distributed random variables with zero mean and unit variance and write \[ \Delta_ n(S)=\sup_{x\in S}| P(\sum^{n}_{i=1}X_ i\leq n^{1/2}x)-\Phi (x)| \] where \(\Phi\) is the standard normal distribution function and \(S\subseteq {\mathbb{R}}\). It is said that S is convergence determining (of order \(n^{-1/2})\) if the ratio \[ (\Delta_ n(S)+n^{- 1/2})/(\Delta_ n({\mathbb{R}})+n^{-1/2}) \] is bounded away from zero and infinity as \(n\to \infty\) for all choices of the distribution of X. In this paper it is shown that no singleton can be convergence determining but any set consisting of four or more distinct points is convergence determining. Doublets or triplets may or may not be convergence determining and a complete characterization of all the convergence determining doublets is provided.