English

This work develops almost sure convergence of negatively associated random vectors in Hilbert spaces. Extensions of a result in [4] are given. Illustrative examples are provided.

This paper establishes mean convergence for the maximum normed partial sums of $$\mathcal{H}$$ -valued random vectors under a general condition of weighted integrability. Our $$L_{p}$$ convergence holds regardless of any dependence structure when $$0<p<1$$ . For $$p=1$$ , the main theorem is applicable to arrays of random vectors exhibiting (i) pairwise and coordinatewise negative dependence, (ii) coordinatewise negative association, and (iii) coordinatewise widely orthant dependence. Notably, the results for cases (i) and (iii) build upon and enhance the main theorem presented by Cabrera and Volodin [J. Math. Anal. Appl. 305, 664–658 (2005)] and Wu et al. [Stochastics 91, 916–944 (2019)], respectively.

A new semiclassical approach to molecular collision dynamics is developed. In this approach the state of a system is described by a unit vector in Hilbert space. By choosing a reference unit vector and a continuous mapping, a correspondence between the vector in Hilbert space and the point in a label space is established, i.e., the vector in Hilbert space is parametrized or labeled by complex variables. It is shown that the label variables formally obey classical mechanics. Thus, the time evolution of the vectors in Hilbert space, i.e., the evolution of the states of the system, can be determined by calculating the evolution of the label variables classically. To illustrate this idea, the formalism for calculating the vibrational transition probability in the collinear collision A+BC is presented, and a demonstrative calculation for the collinear Secrest–Johnson model of He+H2 vibrationally inelastic scattering has been carried out. The comparison with the exact quantum results shows that the agreement is encouragingly good.

We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in R^n, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.

In this study, we consider limit theorems for microscopic stochastic models of neural fields. We show that the Wilson–Cowan equation can be obtained as the limit in uniform convergence on compacts in probability for a sequence of microscopic models when the number of neuron populations distributed in space and the number of neurons per population tend to infinity. This result also allows to obtain limits for qualitatively different stochastic convergence concepts, e.g., convergence in the mean. Further, we present a central limit theorem for the martingale part of the microscopic models which, suitably re-scaled, converges to a centred Gaussian process with independent increments. These two results provide the basis for presenting the neural field Langevin equation, a stochastic differential equation taking values in a Hilbert space, which is the infinite-dimensional analogue of the chemical Langevin equation in the present setting. On a technical level, we apply recently developed law of large numbers and central limit theorems for piecewise deterministic processes taking values in Hilbert spaces to a master equation formulation of stochastic neuronal network models. These theorems are valid for processes taking values in Hilbert spaces, and by this are able to incorporate spatial structures of the underlying model.Mathematics Subject Classification (2000): 60F05, 60J25, 60J75, 92C20.

Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of non-Euclidean data are analyzed to illustrate the usefulness of the proposed general approach.

In this article, we investigate the complete moment convergence for a sequence of coordinatewise pairwise negatively quadrant dependent random vectors in Hilbert space, which is coordinatewise weakly upper bounded by a random vector under special case.

We find the exact weak widths for one class of $2\pi$-periodic functions with values in Hilbert space, which is determined by moduli of continuity.

We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups.

The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument, positive operator valued measure, etc., by using quantum stochastic differential equations and by using classical stochastic differential equations (SDE's) for vectors in Hilbert spaces or for trace-class operators. In the same times Ozawa made developments in the theory of instruments and introduced the related notions of a posteriori states and of information gain [1]. In this paper we introduce a simple class of SDE's relevant to the theory of continual measurements and we recall how they are related to instruments and a posteriori states and, so, to the general formulation of quantum mechanics. Then we introduce and use the notion of information gain and the other results of the paper [1] inside the theory of continual measurements. [1] M. Ozawa, On information gain by quantum measurements of continuous observables, J. Math. Phys. 27 (1986) 759-763.

On complete moment convergence for CAANA random vectors in Hilbert spaces

We develop an approximation theory in Hilbert spaces that generalizes the classical theory of approximation by entire functions of exponential type. The results advance harmonic analysis on manifolds and graphs, thus facilitating data representation, compression, denoising and visualization. These tasks are of great importance to machine learning, complex data analysis and computer vision.KeywordsHilbert SpaceEntire FunctionBesov SpaceExponential TypeSelfadjoint OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Since Pearson's correlation was introduced at the end of the 19th century many dependence measures have appeared in the literature. Recently we have suggested four simple axioms for dependence measures of random variables that take values in Hilbert spaces. We showed that distance correlation satisfies all these axioms. We still need a new measure of dependence because existing measures either do not work in general metric spaces (that are not Hilbert spaces) or they do not satisfy our four simple axioms. The earth mover's correlation introduced in this paper applies in general metric spaces and satisfies our four axioms (two of them in a weaker form).

Some weighted Ostrowski type integral inequalities for vector-valued functions in Hilbert spaces are given. Applications for quadrature rules with values in Hilbert spaces are also pointed out.

In this paper, we obtain the Hájek–Rényi inequality and, as an application, we study the strong law of large numbers for H-valued m-asymptotically almost negatively associated random vectors with mixing coefficients {q(n), ngeq1} such that sum_{n=1}^{infty}q(n)^{2}<infty.