Spectral densities of discrete and continuous-indexed random fields
This text first looks at sequences of discrete-indexed random fields. When these random fields satisfy certain linear dependence conditions uniformly, each will have a spectral density function (not necessarily continuous) that is bounded between two positive constants. These spectral density functions will converge in a weak sense to another function (not necessarily continuous) that is also bounded between two positive constants. Two examples will also be given that show the weak form of convergence seems to be the best one can get. An extra condition on the sequence will also be given which will ensure each spectral density function is continuous and that they uniformly converge to a continuous function.
Continuous-indexed random fields will then be investigated, and linear dependence coefficients specifically for such random fields will be defined. When a selection of these linear dependence conditions are satisfied, the random field will have a continuous spectral density function. Showing this involves the construction of a special class of random fields using a standard Poisson process and the original random field.