A new kind of modified transportation cost inequalities and polynomial concentration inequalities

This paper introduces a new kind of modified transportation cost inequalities and presents some sufficient and necessary conditions for it. Using these conditions, this paper provides some applications of polynomial concentration inequalities to probability measures with no exponential moments. The tensorization property is also provided.     

A refined Jensen’s inequality in Hilbert spaces and empirical approximations

Let f:X→R be a convex mapping and X a Hilbert space. In this paper we prove the following refinement of Jensen’s inequality:

E(f|X∈A)≥E(f|X∈B)     

A multivariate semi-logistic autoregressive process and its characterization

An autoregressive multivariate stochastic model is constructed which yields a stationary Markov process with a marginal invariant distribution as a multivariate semi-logistic distribution. This model is denoted as an MSL-AR(1) process. Some properties of the MSL-AR(1) process are studied and its characterization is also derived.     

Slepian’s inequality with respect to majorization

In this paper, we obtain some sufficient conditions for Slepian’s inequality with respect to majorization for two Gaussian random vectors.     

A new ideal metric with applications to multivariate stable limit theorems

Summary A new ideal metric of orderr>1 is introduced on ^k and a thorough analysis of its metric properties is given. In comparison to the known ideal metric of Zolotarev this new metric allows estimates from above by pseudo difference moments and thus allows applications to stable limit theorems. As applications we give the right order Berry-Esséen type result in the stable case, obtain the limiting behaviour of multivariate summability methods and discuss the approximation problem by compound Poisson distributions.Research supported by NATO GRANT CRG 900 798 and by a DFG Grant     

A concise proof of Kruskal’s theorem on tensor decomposition

A theorem of J. Kruskal from 1977, motivated by a latent-class statistical model, established that under certain explicit conditions the expression of a third-order tensor as the sum of rank-1 tensors is essentially unique. We give a new proof of this fundamental result, which is substantially shorter than both the original one and recent versions along the original lines.     

Hoeffding’s inequality for supermartingales

We give an extension of Hoeffding’s inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.     

A multilinear form inequality

An inequality of Interpolation type for Multilinear Forms with a two-part dependence condition is proved. It generalizes the work of Bradley and Bryc [Theorem 3.6, Multilinear forms and measures of dependence between random variables, J. Multivariate Anal. 16 (1985) 335-367] and Prakasa Rao [Bounds for rth order joint cumulant under rth order strong mixing, Statist. Probab. Lett. 43 (1999) 427-431].     

Some new results on multivariate dispersive ordering of generalized order statistics

Let be generalized order statistics based on a continuous distribution function F with parameters k and (m₁,…,m_n−1). Chen and Hu (2007) [8] investigated the sufficient conditions on F and on the parameters k and m_i’s such that , where , and is the Shaked-Shanthikumar multivariate dispersive order. Since the order does not possess the closure property under marginalization, one may naturally wonder whether the corresponding multivariate margins of the above random vectors are also ordered in the order . This is answered affirmatively in this paper. Some comparison results for generalized order statistics from two samples are presented. Potential applications are also mentioned.     

Cramér large deviation expansions for martingales under Bernstein’s condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.     

Lebesgue’s dominated convergence theorem in Bishop’s style

We present a constructive proof in Bishop’s style of Lebesgue’s dominated convergence theorem in the abstract setting of ordered uniform spaces. The proof generalises to this setting a classical proof in the framework of uniform lattices presented by Hans Weber in [Uniform lattices. II: order continuity and exhaustivity, Annali di Matematica pura ed applicata, (IV) CLXV (1993) 133-158].     

A simple proof of removable singularities for coupled fermion fields

This paper presents a simple proof of a removable singularity theorem for coupled fermion fields on compact four-dimensional manifolds. New methods are employed and the hypotheses here are weak.     

A Zubov’s method for stochastic differential equations

We consider a stochastic differential equation with an asymptotically stable equilibrium point. We show that the domain of attraction of the equilibrium, i.e. the set of points which are attracted with positive probability to it, can be characterized by the solution of a suitable partial differential equation.     

Arveson’s criterion for unitary similarity

This paper is an exposition of W.B. Arveson’s complete invariant for the unitary similarity of complex, irreducible matrices.     

A new dependence ordering with applications

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159-171]. Applications of this partial order to order statistics, k-record values and frailty models are given.     

A class of bivariate exponential distributions

We introduce a class of absolutely continuous bivariate exponential distributions, generated from quadratic forms of standard multivariate normal variates.This class is quite flexible and tractable, since it is regulated by two parameters only, derived from the matrices of the quadratic forms: the correlation and the correlation of the squares of marginal components. A simple representation of the whole class is given in terms of 4-dimensional matrices. Integral forms allow evaluating the distribution function and the density function in most of the cases.The class is introduced as a subclass of bivariate distributions with chi-square marginals; bounds for the dimension of the generating normal variable are underlined in the general case.Finally, we sketch the extension to the multivariate case.     

A new proof of regularity for¶two-shaded image segmentations

In this paper, we consider minimizing the Mumford-Shah functional over two-valued functions in the plane, which is equivalent to minimizing over characteristic functions. Existence of minimizers is straightforward and we show that any minimizing set is essentially open, has a boundary with finitely many connected components, and each component is C ¹. The relatively quick proof does not rely on quasi-minimal surface or varifold theory, but on uniform estimates on the regularity of the boundary. Received: 4 February 1997 / Revised version: 4 February 1998     

Some new results on convolutions of heterogeneous gamma random variables

Convolutions of independent random variables often arise in a natural way in many applied areas. In this paper, we study various stochastic orderings of convolutions of heterogeneous gamma random variables in terms of the majorization order [p-larger order, reciprocal majorization order] of parameter vectors and the likelihood ratio order [dispersive order, hazard rate order, star order, right spread order, mean residual life order] between convolutions of two heterogeneous gamma sets of variables wherein they have both differing scale parameters and differing shape parameters. The results established in this paper strengthen and generalize those known in the literature.