Characterization theorems for some classes of covariance functions associated to vector valued random fields

We characterize some important classes of cross-covariance functions associated to vector valued random fields based on latent dimensions. We also give some results for mixture based models that allow for the construction of new cross-covariance models. In particular, we give a criterion for the permissibility of quasi-arithmetic operators in order to construct valid cross covariances.     

Spectral variation bounds for diagonalisable matrices

Summary This note is related to an earlier paper by Bhatia, Davis, and Kittaneh [4]. For matrices similar to Hermitian, we prove an inequality complementary to the one proved in [4, Theorem 3]. We also disprove a conjecture made in [4] about the norm of a commutator. This work was done when the first author visited the SFB 343 at University of Bielefeld in May and June 1994.     

Irregular sampling and the inverse spectral problem

We are interested in finding necessary and sufficient conditions for irregular sampling to hold. We shall show that the inverse spectral problem can be used to construct sampling type theorems from the knowledge of the sampling points only. This improves Kramer's theorem as it reveals all possible distributions of the sampling points together with a construction of the sampling functions.     

On basicity of exponentials inL P (dμ) and general prediction problems

Let be a nonnegative measure on the unit circle in the complex plane and 1<p<. It is of interest to find conditions on so that the set of exponentialse ⁱⁿ form a strongM-basis forL ^p (d). Some partial results are proved which can shed some light on this important open question. These results are of fundamental importance in the prediction theory of stochastic processes and other fields of applications. These results is then used to obtain a theorem which reduces some prediction problems to easier ones.To 80^th birthday of Paul ErdsThis research is supported by Office of Naval Research Grant No N00014-89-J-1824.     

Cohomologie des groupes localement compacts et produits tensoriels continus de représentations

The following is an expository paper, containing few and sometimes incomplete proofs, on continuous tensor products of Hilbert spaces and of group representations, and on the irreducibility of the latter; the principal results in the last direction are due to Verchik, Gelfand, and Graiev. The theory of continuous tensor products of Hilbert spaces, based on a fundamental theorem of Araki and Woods, is closely related to that of conditionally positive definite functions; it relies on the technique of symmetric Hilbert spaces, which also can be used to give a new proof of the classical Lévy-Khinchin formula (see A. Guichardet, (1973). J. Multiv.3 249–261.). Another basic tool for what follows is the 1-cohomology of unitary representations of locally compact groups; here, the main results are due to P. Delorme; let us mention, for instance, his results for the case of a group G containing a compact subgroup K such that $\text{L}{}^1\text{(Kβ}\text{G}\text{K}\text{)}$ is commutative, using a Lévy-Khinchin's type formula for K-invariant functions due to Gangolli, Faraut, and Harzallah. We add that the results exposed in that paper should have interesting connections with the central limit theorems à la Parthasarathy-Schmidt (see K. Parthasarathy, (1974). J. Multiv. Anal.4 123–149).     

Periodically correlated multivariate second order random distribution fields and stationary cross correlatedness

In this paper we consider multivariate second order operator periodically correlated random distribution fields for which we complete and extend the results from [5].     

Controlling rough paths

We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Hölder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons’ theory of rough paths in Hölder topology.     

On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Some results for fractional boundary value problem of differential inclusions

In this paper, we study fractional differential inclusions with Dirichlet boundary conditions. We prove the existence of a solution under both convexity and nonconvexity conditions on the multi-valued right-hand side. The proofs rely on nonlinear alternative Leray–Schauder type, Bressan–Colombo selection theorem and Covitz and Nadler’s fixed point theorem for multi-valued contractions. The compactness of the set solutions and relaxation results is also established. In the last section we consider the fractional boundary value problem with infinite delay.     

Continuous version of Filippov’s theorem for fractional differential inclusions

Using Bressan-Colombo results, concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values, we prove a continuous version of Filippov’s theorem for a fractional differential inclusion.     

On Favard's theorems

In this paper, we improve and extend the classical Favard's theorems, i.e. Favard's theorem of the module containment, Favard's theorem of linear differential equations. We study Favard's theory of linear differential equations with piecewise constant argument. An example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument. The equivalence between almost automorphic functions and N-almost periodic ones is studied.     

Dimension and entropy of a non-ergodic Markovian process and its relation to Rademacher Riesz Products

Given a Markov chain (not necessarily stationary or homogeneous) with finite state space and an initial distribution, we can construct a measure on the unit interval [0, 1]. In this work we examine the equality (up to a constant) of the Hausdorff dimension of and of a suitably defined entropy for the Markovian process. The results are applied to the so-called Rademacher-Riesz Products.     

Construction of determinantal representation of trigonometric polynomials

For a pair of n×n Hermitian matrices H and K, a real ternary homogeneous polynomial defined by F(t,x,y)=det(tI_n+xH+yK) is hyperbolic with respect to (1,0,0). The Fiedler conjecture (or Lax conjecture) is recently affirmed, namely, for any real ternary hyperbolic polynomial F(t,x,y), there exist real symmetric matrices S₁ and S₂ such that F(t,x,y)=det(tI_n+xS₁+yS₂). In this paper, we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials.     

Interpolation, correlation identities, and inequalities for infinitely divisible variables

We present an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables. This is then applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities. Acknowledgements and Notes. The research of C. Houdré was supported in part by an NSF Mathematical Sciences Post-Doctoral Fellowship and by an NSF-NATO Postdoctoral Fellowship and by the NSF grant No. DMS-98032039. This research was completed while V. Pérez-Abreu was visiting the Georgia Institute of Technology.     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.     

Wavelet analysis and covariance structure of some classes of non-stationary processes

Processes with stationary n-increments are known to be characterized by the stationarity of their continuous wavelet coefficients. We extend this result to the case of processes with stationary fractional increments and locally stationary processes. Then we give two applications of these properties. First, we derive the explicit covariance structure of processes with stationary n-increments. Second, for fractional Brownian motion, the stationarity of the fractional increments of order greater than the Hurst exponent is recovered.     

Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field's canonical metric; the full force of Fernique's zero-one laws and Talagrand's theory of majorizing measures is required. The second characterization ties the modulus to the field's random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation's additive noise; a random Fourier series characterization is also given.