Several aspects of the mean square estimation of a second-order random variable based upon elements from a random field are considered. The role of the underlying probability space is stressed. Numerous examples are presented that point out many of the subtleties associated with this endeavor.>

For any positive integer k and any integer N > 1, a probability space, a Gaussian vector X taking values in R/sup 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.>

It is shown that multidimensional convolution need not be associative. Further, for any positive integer k, it is shown that the multidimensional convolution of two real valued, bounded, integrable, nowhere zero functions defined in R/sup k/ can be identically equal to zero. These results are discussed in an algebraic setting, and a consequence involving random fields is briefly considered. Several aspects of multidimensional convolution that would be of interest to the signal processing community are included.>

The authors consider the problem of optimal estimation of a random variable X based on an observation denoted by a random vector Y. A commonly encountered problem involves estimating X via h(Y) so as to minimize E( Phi (X-h(Y))), where h is Borel measurable and Phi is a Borel measurable cost function chosen to adequately reflect the fidelity demands of the problem under consideration. The authors place a mild condition on the regular conditional distribution of X given sigma (Y) that ensures that E( Phi (X-h(Y))) is minimized for any cost function Phi that is nonnegative, even and convex. In addition, it is shown that given any Borel measurable function g: R to R, there exist random variables X and Y possessing a joint density function such that E(X mod Y=y)=g(y) almost everywhere with respect to Lebesgue measure.>

The problem of fusing or combining various estimates to obtain a single good estimate is investigated. The problem of fusion in estimation theory is addressed, and several examples using common distributions in which virtually any method of fusion would be useless in approximating the random variable of interest are presented. A theorem which, for a very general situation, shows that fusion resulting in an almost surely exact approximation is always possible is presented. In particular, this result addresses the situation in which the data consists of the random variable of interest corrupted by additive Gaussian noise and the random variable of interest could be any second-order random variable. Finally, an example which illustrates the utility of this result is presented.>