Complex random fields

Abstract The fundamental properties of complex random fields are derived directly in an n-dimensional setting and are not inferred as generalizations of the one-dimensional case. In particular, fields with orthogonal increments and stochastic integrals with respect to such fields are defined and their elementary properties analyzed. The spectral representation theorem for homogeneous fields is proved, and various second order properties resulting from the application of linear difference and differential operators to such fields are deduced. The specialization to isotropic fields is considered. Finally, white fields are defined and their characteristic property exploited.