No abstract

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 Matérn 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 Matérn model.

Existence 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.

The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.

ABSTRACT. This paper gives necessary and sufficient conditions for an invariant subspace Jt of H2(T2) to be of the form qH2(T2) (q inner) in terms of double commutativity of the shifts. Recent results in [8] follow directly from our work. Relation to the work in [1] is also discussed.In [5], Rudin gave an example of a shift-invariant subspace of H2(T2) which is not of the form qH2(T2), where q is an inner function. In addition, [5] gives an example of an "outer function" / for which ^f, the smallest invariant subspace generated by / is not equal to H2(T2). Motivated from the prediction theory for random fields, Soltani [8] gave complicated necessary and sufficient conditions for an "outer function" in the sense of [5] to satisfy ./#/■ = H2(T2). Our purpose here is to use the Theorem of Halmos for two commuting isometries due to Slocinski [7] to obtain necessary and sufficient conditions on the invariant subspace ^# of H2(T2) to be of the form qH2(T2) (q inner). As a consequence of this result we obtain necessary and sufficient conditions on jfá¡ for an outer function /, to be H2(T2). We use results of [3] to relate our conditions to those in [8]. As a by-product of our main theorem we show that all invariant subspaces unitarily equivalent to qH2(T2) are exactly of the same form. The last result relates to some recent work of Agrawal, Clark, and Douglas [1], where the problem of unitary equivalence of invariant subspaces of H2(Tn) was solved under some sufficient conditions. Their case does not include subspaces of the type considered here. We thank the referee and the editor for bringing the work in [1] to our attention. We begin with some notation.Let Z be the set of integers. We denote by m,n etc. the elements of Z. Let U be the open unit disc and T the boundary of U in the complex plane C. Let Z2,C2, [/2 and T2 be the respective cartesian products and a2 the normalized Lebesgue measure on T2. For p > 0, we denote by Lp(T2,a2) the usual Lebesgue space of the equivalence class of p-integrable functions and HP(U2) ={/:/:U2 -► C analytic and sup f |/r(t)|pda2 < ooHere /r(t) = f(z) with z = rt. Let z = (zuz2) = (nei01,r2eie2) and t = (ei6l,ei02), then P(z,t) = Pri(0i -62). Pr2(6i -62) is called Poisson kernel with