Models for the difference of continuous covariance functions

Abstract A linear combination, with negative weights, of two continuous covariance functions has been analyzed by a few authors just for special cases and only in the real domain. However, a covariance is a complex valued function: for this reason, the general problem concerning the difference of two covariance functions in the complex domain needs to be analyzed, while it does not yet seem to have been addressed in the literature; hence, exploring the conditions such that the difference of two covariance functions is again a covariance function can be considered a further property. Therefore, this paper yields a contribution to the theory of correlation, hence the results cannot be restricted to the particular field of application. Starting from the difference of two complex covariance functions defined in one dimensional Euclidean space, wide families of models for the difference of two complex covariance functions can be built in any dimensional space, utilizing some well known properties. In particular, the difference of two real covariance functions has been considered; moreover, the difference between some special isotropic covariance functions has also been analyzed. A detailed analysis of the parameters of the models involved has been proposed; this kind of analysis opens a gate for modeling, in any dimensional space, the correlation structure of a peculiar class of complex valued random fields, as well as the subset of real valued random fields. Some relevant hints about how to construct the subset of real covariance functions characterized by negative values have also been given.