On characteristic functions of multidimensional distributions

We consider the possibility of a multidimensional generalization of M. Krein’s theorem on extension of a positive-definite function from an interval to the real line with preservation of positive definiteness. Under some additional conditions, this problem is reduced to the lattice case. An extension criterion for the lattice case is given. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 143–149. Translated by S. Yu. Pilyugin.     

Fourier inversion for multidimensional characteristic functions

A density functionf(x),xR ⁿ is said to bepiecewise smooth if for eachxR ⁿ , the mean value function is piecewiseC with compact support. (d is normalized surface measure on the unit sphere). The Fourier transform is with spherical partial sum . Theorem. For suchf, lim _r f _R (x)=M ₀+f(x) if and only ifrM _r f(x) hask=[(n–3)/2] continuous derivatives. ([]=integer part). Otherwise we have lim where 0 is uniquely determined.     

Some remarks on spherically invariant distributions

We consider the notion of a spherically invariant measure on a real and separable Hilbert space and study some equivalence properties of measures of this type. The latter are of interest in statistical communication theory.     

An upper bound for Hilbert cubes

In this note we give a new upper bound for the largest size of subset of {1,2,…,n} not containing a k-cube.     

改进型Hardy-Sobolev不等式最好常数的一个上界

在本文,我们对改进型Hardy-Sobolev不等式的最好常数进行研究,得到该常数的一个上界.     

On the finite series expansion of multivariate characteristic functions

Three theorems are obtained that relate the asymptotic behavior of a distribution function with the behavior of its characteristic function at the origin. These theorems generalize one dimensional results that have been obtained by the author and by others.     

Capturing the multivariate extremal index: bounds and interconnections

The multivariate extremal index function is a direction specific extension of the well-known univariate extremal index. Since statistical inference on this function is difficult it is desirable to have a broad characterization of its attributes. We extend the set of common properties of the multivariate extremal index function and derive sharp bounds for the entire function given only marginal dependence. Our results correspond to certain restrictions on the two dependence functions defining the multivariate extremal index, which are opposed to Smith and Weissman’s (1996) conjecture on arbitrary dependence functions. We show further how another popular dependence measure, the extremal coefficient, is closely related to the multivariate extremal index. Thus, given the value of the former it turns out that the above bounds may be improved substantially. Conversely, we specify improved bounds for the extremal coefficient itself that capitalize on marginal dependence, thereby approximating two views of dependence that have frequently been treated separately. Our results are completed with example processes.      

Some inequalities for characteristic functions

In a recent paper, Matysiak and Szablowski [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] posed an interesting conjecture about a lower bound of real-valued characteristic functions. Under a suitable moment condition on distributions, we prove the conjecture to be true. The unified approach proposed here enables us to obtain new inequalities for characteristic functions. We also show by example that the improvement in the bounds is significant if more information about the distribution is available.     

Some inequalities for characteristic functions. II

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] about a lower bound of real-valued characteristic functions. In this paper, applying an alternative approach we are able to give explicitly the ranges of argument for which the obtained inequalities hold true for general characteristic functions.     

On some integral properties of multidimensional Hilbert operators

In the present paper, we study the integral properties of multidimensional Hilbert transforms.     

Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems

Assume that K⊂Rnm is a convex body with o∈int(K) and is a function with f|K∈C⁰(K,R) and f|(Rnm?K)≡+∞. We show that its lower semicontinuous quasiconvex envelope

     

Analog of heat equation for gaussian measure of a ball in Hilberrt space

If _a,T is a Gaussian measure on a Hilbert space with meana and covariance operatorT, andr is a fixed positive number, then the functiong(a,T)= _a,T {xr} satisfies the equation

Products of generalized Nevanlinna functions with symmetric rational functions

New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized Nevanlinna function in a constructive manner. Also, a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function. The operator theoretic treatment of these realizations is based on the notions of rigged spaces, boundary triplets, and associated Weyl functions.     

A multivariate empirical characteristic function test of independence with normal marginals

This paper proposes a semi-parametric test of independence (or serial independence) between marginal vectors each of which is normally distributed but without assuming the joint normality of these marginal vectors. The test statistic is a Cramér–von Mises functional of a process defined from the empirical characteristic function. This process is defined similarly as the process of Ghoudi et al. [J. Multivariate Anal. 79 (2001) 191] built from the empirical distribution function and used to test for independence between univariate marginal variables. The test statistic can be represented as a V-statistic. It is consistent to detect any form of dependence. The weak convergence of the process is derived. The asymptotic distribution of the Cramér–von Mises functionals is approximated by the Cornish–Fisher expansion using a recursive formula for cumulants and inversion of the characteristic function with numerical evaluation of the eigenvalues. The test statistic is finally compared with Wilks statistic for testing the parametric hypothesis of independence in the one-way MANOVA model with random effects.     

Gap Functions and Error Bounds for Weak Generalized Ky Fan Inequalities

In this paper, we study a weak generalized Ky Fan inequality with cone constraints through image space analysis. First, we characterize the separation for the weak generalized Ky Fan inequality with cone constraints using the saddle points of generalized Lagrangian function. Then, we use regular weak separation functions to construct gap functions and regularized gap functions for the weak generalized Ky Fan inequality with cone constraints in a general way, and establish its error bounds in terms of these gap functions.