Central limit theorems for weighted sums of exchangeable random elements in banach spaces

A generalization of a central limit theorem for martingale difference arrays due to D. L. McLeish is obtained for random elements in a separable Banach space E. This result, with a technique of N. C. Weber, is used to obtain a weak convergence theorem for weighted sums of a row-wise exchangeable array (X nk ) of random elements in a Banach space E which is uniformly 2-smooth. Corollaries include a central limit theorem for weighted sums of an exchangeable sequence (X k ), and several weak laws of large numbers for such weighted sums