Necessary and sufficient conditions for the classical central limit theorem (CLT) for i.i.d. random vectors in an arbitrary separable Banach space require not only assumptions on the original distribution, but also on the sample. What we do here is to continue our study of the CLT in terms of the original distribution. Of course, some new ingredient must be introduced, so we allow slight modifications of the random vectors. In particular, we restrict our modifications to be continuous, and to be no larger than a fixed small number, or in some cases a fixed small proportion of the magnitude of the individual elements of the sample. We find that if we use certain interpolation space norms to measure the magnitude of such modifications, then the CLT can be improved. Examples of our result are also included.
Digital Object Identifier: 10.1214/09-IMSCOLL506