V. Mandrekar

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 Matern 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 Matern model.

The validity of Beurling theorems in polydiscs

Abstract : Let Z be the set of integers. We denote by m,n etc. the elements of Z sub 2. Let U sub 2 be the open unit disc and T the boundary of U in the complex plane C slashed. Let Z sub 2, U sub 2 and T sub 2 be the respective calesian product and delta sub 2 the normalized Lebesgue measure on T sub 2. For p0, we denote by L sub p (T sub 2, delta sub 2) the normalized Lebesgue space of the equivalence class of p-integrable functions. (Author)

Stochastic Differential Equations in Infinite Dimensions

Preface.- Part I: Stochastic Differential Equations in Infinite Dimensions.- 1.Partial Differential Equations as Equations in Infinite.- 2.Stochastic Calculus.- 3.Stochastic Differential Equations.- 4.Solutions by Variational Method.- 5.Stochastic Differential Equations with Discontinuous Drift.- Part II: Stability, Boundedness, and Invariant Measures.- 6.Stability Theory for Strong and Mild Solutions.- 7.Ultimate Boundedness and Invariant Measure.- References.- Index.

Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces

The stochastic integrals of M- type 2 Banach valued random functions w.r.t. compensated Poisson random measures introduced in (Rüdiger, B., 2004, In: Stoch. Stoch. Rep., 76, 213–242.) are discussed for general random functions. These are used to solve stochastic integral equations driven by non Gaussian Lévy noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local Lipshitz conditions for the drift and noise coefficients on M-type 2 as well as general separable Banach spaces. The continuous dependence of the solution on the initial data as well as on the drift and noise coefficients are shown. The Markov properties for the solutions are analyzed.

Equivalence and singularity of Gaussian measures and applications

Keywords: reproducing kernel Hilbert spaces;;; equivalence and singularity;;; Gaussian measures;;; expository paper;;; Feldman-Hajek dichotomy for Gaussian measures;;; stationary Gaussian processes;;; absolute continuity and singularity of probability measures Reference GPRO-CHAPTER-1978-003 Record created on 2010-05-25, modified on 2016-08-08

The spectral representation of stable processes: Harmonizability and regularity

SummaryWe show that symmetric α-stable moving average processes are not harmonizable. However, we show that a concept of generalized spectrum holds for allLp-bounded processes O<p<-2. In capep=2, generalized spectrum is a measure and the classical representation follows. For strongly harmonizable symmetric α-stable processes we derive necessary and sufficient conditions for the regularity and the singularity for 0<α≦2, using known results on the invariant subspaces. We also get Cramér-Wold decomposition for the case 0<α≦2.

Spectral theory of stationary H-valued processes

For weakly stationary stochastic processes taking values in a Hilbert space, spectral representation and Cramer 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.

Quasi-invariance of measures under translation

Let # be a probability measure on the Bore1 field Z of the real line IR and let P = # | 1 7 4 be the product measure defined on the Borel field 2 ; ~ 1 7 6 1 7 4 1 7 4 -of the topological vector space I R ~ = I R | 1 7 4 .-. . If a = ( a , ) , e I is an element of IR ~, let P~ be the measure P translated by a i.e. P,,(A)=P(A+a) for A s Z ~. In the present paper we study the set E(P)= {aslR~~ where P ~ P , signifies that P ( A ) = 0 iff P,(A)=0. Previously E(P) had been studied by, amongst others, Dudley [4], Feldman [5], Shepp [9]. Our main result (w 3) completes theirs by showing that E(P) contains a Mazur-Orlicz vector subspace 1 ~ (for definitions see w 2) whenever it contains some 1-dimensional vector subspace and that whenever E(P) is a vector subspace then E(P)= l ~ We also give an explicit expression for the function 0. A counter-example based on one due to Dudley given in the appendix shows that E(P) need not always be a vector subspace of IR ~~ However, in w 4, we give a simple sufficient condition for E(P) to be a subspace and exhibit wide classes of examples of E(P)= l ~ for various functions 0. Our results were announced previously in [2] without proofs.

On singularity and Lebesgue type decomposition for operator-valued measures

The concepts of absolute continuity and singularity for operator-valued measures are introduced and Radon-Nikodym and Lebesgue decomposition theorems for such measures are established. These theorems reduce directly to the classical results in the scalar case. The results have interesting applications to the theory of infinite-dimensional stationary stochastic processes.

Harmonic analysis on certain vector spaces

Abstract : Let 1 denote the vector space of all sequences of real numbers with the topology of coordinate-wise convergence. For 0 < p < infinity l sub p denote the subset of l consisting of all sequences x sub i which have the summation, from one to infinity of the (absolute value of x sub i) to the p power finite. The main efforts in the paper are to generalize Bochners theorem and the Levy-continuity theorem to these l sub p spaces. (Author)