Fixed-domain asymptotic properties of tapered maximum likelihood estimators

When the spatial sample size is extremely large, which occurs in many environmental and ecological studies, operations on the large covariance matrix are a numerical challenge. Covariance tapering is a technique to alleviate the numerical challenges. Under the assumption that data are collected along a line in a bounded region, we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator (MLE) for the microergodic parameter in the Mat\'ern covariance function by establishing the fixed-domain asymptotic distribution of the exact MLE and that of the tapered MLE. Our results imply that, under some conditions on the taper, the tapered MLE is asymptotically as efficient as the true MLE for the microergodic parameter in the Mat\'ern model.Comment: Published in at http://dx.doi.org/10.1214/08-AOS676 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spectral theory of stationary H-valued processes

AbstractFor weakly stationary stochastic processes taking values in a Hilbert space, spectral representation and Cramér decomposition are studied. Using these ideas and the moving average representation for such processes established earlier by the authors, some necessary and sufficient spectral conditions for such stochastic processes to be purely nondeterministic are given in both discrete and continuous parameter cases

On the existence of weak variational solutions to stochastic differential equations

We study the existence of weak variational solutions in a Gelfand triplet of real separable Hilbert spaces, under continuity, growth, and coercivity conditions on the coefficients of the stochastic differential equation. The laws of finite dimensional approximations are proved to weakly converge to the limit which is identified as a weak solution. The solution is an H– valued continuous process in L2 (Ω, C([0, T], H)) ∩ L2([0, T] × Ω, V ). Under the assumption of monotonicity the solution is strong and unique

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

AbstractExistence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai

Applications of Multiplicity Theory to N-ple Markov Processes

1 online resource (PDF, 26 pages

On Multivariate Wide-Sense Markov Processes

1 online resource (PDF, 12 pages