Uniform and nonuniform estimates in the CLT for Banach valued dependent random variables

A uniform estimate of the rate of convergence in the central limit theorem (CLT) in certain Banach spaces for dependent random variables is established when the Gaussian measure of the [epsilon]-neighbourhood of the boundary of a set is proportional to [epsilon] and the third order moment is finite in the strong sense. A uniform estimate in the CLT for Banach valued dependent random variables is carried out when the B-space is well behaved for a martingale transform.