On the central limit theorem in Rp when p→∞

Abstract

Let X1, X2, ..., Xn be i.i.d. random vectors in Rp where p tends to infinity. A theorem is presented showing that the Central Limit Theorem should hold if p2/n tends to zero. Furthermore, an example is presented with Xi having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean {Mathematical expression} can not hold if p2/n does not tend to zero. © 1986 Springer-Verlag.