Krein factorization of covariance operators of 2-parameter random fields and application to the likelihood ratio

A factorization of the covariance operator (I+R) is derived for the observation process of a 2-parameter random field. This result can be applied to express the determinant term appearing in L.A. Shepp´s (1966) expression for the likelihood ratio in terms of the system parameters. This means that, in practice, one of the problems in computing the likelihood ratio for random fields is solved. Extensions to the multiparameter case are straightforward. The expression of the determinant of (I+R) in terms of the system parameters may also be used to reexpress the Wong-Zakai correction term (1977)