Edgeworth series expansion of the conditional mean and the optimality of non-linear Volterra filters

The authors address the problem of the optimality of the Volterra filters, in comparison with the optimal estimator in a minimum mean square error sense: the conditional mean. Notation and definition concerning cumulants, moments, generalized cumulants and Hermite tensors are introduced. The best Volterra estimation of a random variable x from a random observation vector Y is calculated, and an Edgeworth series expansion of the conditional mean, the optimal estimator, is given. In the case of a Gaussian observation vector, it is shown that the Volterra estimator and the expansion of the conditional mean are equal, providing a result concerning the optimality of Volterra filters