Weaker conditions for innovations informational equivalence in the independent Gaussian case

This paper deals with the problem of establishing conditions under which, in the independent Gaussian case, a stochastic process can be considered to be informationally equivalent to its innovations. In recent years this problem has been considered and, as a result, sufficient conditions implying informational equivalence are now available. On the other hand, these conditions are stronger than the ones implying whiteness of the innovations process. The aim of this paper is to fill the gap between conditions assuring whiteness of the innovations and the ones implying informational equivalence. More specifically, by considering the innovations problem in the context of multiplicity theory of stochastic processes and using the notion of a fully submitted process, a necessary and sufficient condition for informational equivalence of the innovations is established. This condition can be interpreted as a condition of nonsingularity in detection theory and turns out to be weaker than the measure-theoretic equivalence condition that has been used in essentially all of the most recent contributions to the innovations problem.