Abstract
We examine the equivalence and orthogonality of two zero-mean Gaussian measures on the sigma-field generated by a stationary random function sampled on a subset of a real coordinate space. We give the isotropic analog to the result that equivalent Gaussian measures are linked with the existence of a square-integrable extension of the difference between the covariance functions from the sampling domain to the entire space. This is done using the two-dimensional Hankel transform and general methods for Gaussian random functions. Our results allow us to further expand the methodology for identifying orthogonal Gaussian measures. In particular, we provide means to obtain orthogonal Gaussian measures if the sampling domain is dense in the entire space. As a natural complement to this, we obtain orthogonal measures based on the assumption that the set of distances from points of the sampling domain to the origin is dense in the set of non-negative real numbers. The latter finding provides means to deduce the orthogonality of Gaussian measures when the random function is sampled along a continuous and unbounded path taking values in a real coordinate space.
In terms of statistical inference on random functions, we relate on methods to deduce the consistency of maximum likelihood (ML) covariance parameter estimators. To this end, we focus on two methods: The first revolves around the martingale family defined upon the Radon–Nikodym derivative obtained from the finite dimensional distribution of the random function and the second is subject to the condition that the minimum distance between coordinates from the sampling domain is uniformly bounded away from zero.
Regarding the first method, we elaborate on the relationship between orthogonal distributions, the uniform convergence of the latter martingale family and the (strong) consistency of ML-estimators. We point out a relationship with the assumption that an instance of the conditional entropy associated with the sigma-field generated by the martingale filtration is uniformly bounded. In particular, we study the latter assumption on the conditional entropy in the context of measurable partitions of Lebesgue spaces. As a main result, we provide conditions under which the sigma-field generated by the martingale filtration conveys too much information for the maximal conditional entropy to be finite – our arguments are accompanied by several classic examples of function spaces.
Referring to the second method, our results are inspired by the work of Bachoc (2014). If the sampling domain is given by a perturbed regular grid, we give conditions to deduce the (weak) consistency of ML-estimators in the context of compactly supported covariance functions. As an extension, we introduce the notion of truncated-ML-estimators – estimators which are based on approximations of the presumed family of covariance functions. If the latter family belongs to a family of compactly supported covariance functions, we provide conditions to recover the (weak) consistency for truncated-ML-estimators.
Hediger, Michael