Central limit theorems for weighted sums of exchangeable random elements in banach spaces ⁽¹⁾

⁽¹⁾Research supported by a grant form the Vanderbilt University Research Council.

Abstract

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

⁽¹⁾Research supported by a grant form the Vanderbilt University Research Council.

⁽¹⁾Research supported by a grant form the Vanderbilt University Research Council.

Notes

⁽¹⁾Research supported by a grant form the Vanderbilt University Research Council.