On the central limit theorem in \(R^ p\) when p\(\rightarrow \infty\) (Q1071369)

Let \(X_ 1,X_ 2,...,X_ n\) be i.i.d. random vectors in \(R^ p\) where p tends to infinity. A theorem is presented showing that the central limit theorem should hold if \(p^ 2/n\) tends to zero. Furthermore, an example is presented with \(X_ i\) having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean (\(\sqrt{n}\bar X)\) can not hold if \(p^ 2/n\) does not tend to zero.