Some properties of a class of symmetric functions and its applications

For , the symmetric functions are defined by where , and are non‐negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n‐dimensional space and others.     

Strict inequalities for the derivatives of functions satisfying certain boundary conditions

For functions satisfying the boundary conditions

Lagrangian function and duality theory in multiobjective programming with set functions

Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions,D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex coneD. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.The authors greatly appreciate helpful and valuable comments and suggestions received from the referee.     

Spaces of integrable functions with respect to a vector measure and factorizations through and Hilbert spaces

We use the integration structure of the spaces of scalar integrable functions with respect to a vector measure to provide factorization theorems for operators between Banach function spaces through Hilbert spaces. A broad class of Banach function spaces can be represented as spaces of scalar integrable functions with respect to a vector measure, but this representation (the vector measure) is not unique. Since our factorization depends on the vector measure that is used for the representation we also give a characterization of those vector measures whose corresponding spaces of integrable functions coincide.     

Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.     

Some properties of invariant polynomials with matrix arguments and their applications in econometrics

Summary Further properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes thek-class estimator in econometrics.     

A Formula for the Tail Probability of a Multivariate Normal Distribution and Its Applications

An exact asymptotic formula for the tail probability of a multivariate normal distribution is derived. This formula is applied to establish two asymptotic results for the maximum deviation from the mean: the weak convergence to the Gumbel distribution of a normalized maximum deviation and the precise almost sure rate of growth of the maximum deviation. The latter result gives rise to a diagnostic tool for checking multivariate normality by a simple graph in the plane. Some simulation results are presented.     

Vector measure Maurey–Rosenthal-type factorizations and -sums of -spaces

Consider a vector measure of bounded variation m with values in a Banach space and an operator T:XL¹(m), where L¹(m) is the space of integrable functions with respect to m. We characterize when T can be factorized through the space L²(m) by means of a multiplication operator given by a function of L²(|m|), where |m| is the variation of m, extending in this way the Maurey–Rosenthal Theorem. We use this result to obtain information about the structure of the space L¹(m) when m is a sequential vector measure. In this case the space L¹(m) is an ℓ-sum of L¹-spaces.     

The empirical distribution function and strong laws for functions of order statistics of uniform spacings

Let N points be arbitrarily chosen on the circle with unit circumference, and order them clockwise. The uniform mth order spacings are then defined as the clockwise distances between any pair of points having m − 1 other points in between. A Glivenko-Cantelli theorem and nonlinear almost sure bounds for the empirical distribution function based on these uniform spacings are derived. The parameter m is allowed to increase with N to infinity. Applications to linear combinations of functions of mth order spacings are given.     

On weighted approximations in D[0, 1] with applications to self-normalized partial sum processes

Let X,X ₁,X ₂,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S _[nt], 0 ≦ t ≦ 1} where S _n = Σ _j=1 ⁿ X _j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S _[nt]=V _n , 0 ≦ t ≦ 1}, where V _n ² = Σ _j=1 ⁿ X _j ² . L _p approximations of self-normalized partial sum processes are also discussed.     

Semicontinuity and Supremal Representation in the Calculus of Variations

We study the weak* lower semicontinuity properties of functionals of the form

Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences

Recently, Fishbum and Lavalle (1995) and Lefèvre and Utev (1996) have considered some stochastic order relations specific for arithmetic random variables. The present work is concerned with these orderings, together with two other classes of stochastic order relations closely related. First, attention is paid to characterizations and various properties of all these orderings. Then, sufficient conditions of crossing-type for the two new classes of orderings are derived and extrema among discrete random variables are deduced. This is applied in actuarial sciences to obtain new bounds for the classical single life premiums as well as for the probability of ruin in the compound binomial risk model.     

Banzhaf Measures for Games with Several Levels of Approval in the Input and Output

An axiomatic characterization of ‘a Banzhaf score’ notion is provided for a class of games called (j,k) simple games with a numeric measure associated to the output set, i.e., games with n players, j ordered qualitative alternatives in the input level and k possible ordered quantitative alternatives in the output. Three Banzhaf measures are also introduced which can be used to determine a player's ‘a priori’ value in such a game. We illustrate by means of several real world examples how to compute these measures. Research partially supported by Grant BFM 2003-01314 of the Science and Technology Spanish Ministry and the European Regional Development Fund.     

Köthe spaces and topological algebra with bases

Nuclear Köthe sequence spaceλ(P) its crossdualλ(P) ^x and their non-nuclear variants are examined as topological algebras. Modelling on them, a general theory of nuclear topological algebras with orthogonal basis is developed. As a by-product, abstract characterizations of sequence algebras ?^∞ andc ₀ are obtained. In a topological algebra set-up, an abstract Grothendieck-Pietsch nuclearity criterion is developed.     

Minimal cyclic random motion in and hyper-Bessel functions

We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocity c in . The n+1 possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with time t), the geometrical features of which are here investigated.The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of order n+1. These distributions are proved to be connected with (n+1)th order p.d.e. which can be reduced to Bessel equations of higher order.Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [E. Orsingher, A.M. Sommella, A cyclic random motion in R³ with four directions and finite velocity, Stochastics Stochastics Rep. 76 (2) (2004) 113–133] which is here proved to be false.     

Cone-valued Lyapunov functions and stability for impulsive functional differential equations

The stability criteria in terms of two measures for impulsive functional differential equations are established via cone-valued Lyapunov functions and Razumikhin technique. The stability can be deduced from the (Q₀,Q)-stability of comparison impulsive differential equations. An example is given to illustrate the advantages of the results obtained.     

Bounded universal functions in one and several complex variables

We show how to obtain functions that are universal for the ball of , where . The existence of our functions will follow from universality criteria, but we also show how to construct them. Then we study the connection between certain interpolating sequences, runaway automorphisms, and the existence of universal functions on domains in .      

Chaos in Kolmogorov systems with proliferation - general criteria and applications

We provide general criteria for existence of chaotic dynamics in Kolmogorov systems with proliferation. As applications, we provide a new proof of chaoticity of a class of constant coefficient birth-and-death systems and also show that a finite difference discretization of a class of drift diffusion equations generates chaotic dynamics. We also present a general method of constructing chaotic Kolmogorov systems.     

Archimedean copulas in finite and infinite dimensions—with application to ruin problems

In this paper we discuss the link between Archimedean copulas and L₁ Dirichlet distributions for both finite and infinite dimensions. With motivation from the recent papers Weng et al. (2009) and Albrecher et al. (2011) we apply our results to certain ruin problems.