On a failure of multidimensional quantization

For any positive integer k and any integer N > 1, a probability space, a Gaussian vector X taking values in R^k defined on the space with a positive definite covariance matrix, and an N-level k-dimensional quantizer Q are presented such that the random vector Q(X) takes on each of the N values in its range with equal probability and such that X and Q(X) are independent. It is pointed out that the results obtained can be used to answer many questions which arise when quantization is employed. Examples of this are provided