Variance inequalities for covariance kernels and applications to central limit theorems

A simple estimate for the error in the CLT, valid for a wide class of absolutely continuous r.v.'s, is derived without Fourier techniques. This is achieved by using a simple convolution inequality for the variance of covariance kernels or w-functions in conjunction with bounds for the total variation distance. The results are extended to the multivariate case. Finally, a simple proof of the classical Darmois–Skitovich characterization of normality is obtained.