Inequalities for E k(X,Y) when the marginals are fixed

When k(x, y) is a quasi-monotone function and the random variables X and Y have fixed distributions, it is shown under some further mild conditions that k(X, Y) is a monotone functional of the joint distribution function of X and Y. Its infimum and supremum are both attained and correspond to explicitly described joint distribution functions.Research supported by the Air Force Office of Scientific Research under Grant AFOSR-75-2796Research supported by the National Science Foundation     

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.     

A note on the gl constant ofE \mathop \otimes \limits^ \vee F F

LetX andY be Banach spaces. TFAE (1)X andY do not contain subspaces uniformly isomorphic to (2) The local unconditional structure constant of the space of bounded operatorsL (X*_k,Y _k) tends to infinity for every increasing sequence and of finite-dimensional subspaces ofX andY respectively.     

Two-dimensional Meixner Random Vectors of Class ${\mathcal{M}}_{L}$

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called “non-degeneracy”.     

The limit space Cc(X,Y) and the connected components of X and Y

Let X and Y be limit spaces (in the sense of FISCHER). For f ? C(X, Y), let [f] denote the subset of C(X, Y), where the maps take the connected components of X into those of Y quite analogously to f. The subspace [f] of the continuous convergence space C_c(X, Y) is written as a product II C_c(X_i, Y_k(i)), where X_i runs through the components of X and Y_k(i) always is the component of Y which contains the set f(X_i). Sufficient conditions for the representation C_c(X, Y) = Σ [f] are given (in terms of the spaces X and Y). Some applications on limit homeomorphism groups are included.     

Point and Interval Estimation of P(X < Y): The Normal Case with Common Coefficient of Variation

The problem of estimating R = P(X < Y) originated in the context of reliability where Y represents the strength subjected to a stress X. In this paper we consider the problem of estimating R when X and Y have independent normal distributions with equal coefficient of variation. The maximum likelihood estimation of R when the coefficient of variation is known and when it is unknown is studied. The asymptotic variance of the estimators are obtained and asymptotic confidence intervals are provided. An example is presented to illustrate the procedure. Finally some simulation studies are carried out to study the coverage probability and the lengths of the confidence interval. In particular, lengths of the confidence intervals are compared with and without the assumption of common coefficient of variation. It is observed that the assumption of common coefficient of variation results in considerably tighter intervals.     

Tail asymptotics under beta random scaling

Let X,Y,B be three independent random variables such that X has the same distribution function as YB. Assume that B is a beta random variable with positive parameters α,β and Y has distribution function H with H(0)=0. In this paper we derive a recursive formula for calculation of H, if the distribution function Hα,β of X is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y, which is closely related to asymptotics of Weyl fractional-order integral operators. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and Hα,β, respectively, and the conditional limiting distribution of bivariate elliptical distributions.     

A characteristic property of the exponential distribution

The following characterization of the exponential distribution is given: Under suitable conditions on the random variables X and Y, X is exponentially distributed if and only if E[min{X, Y}]=E(X)P(X<Y).     

Positive definite solution of the matrix equation \boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}

Consider the nonlinear matrix equation X?=?Q?+?A ^H (I???X???C) ^?? A ( ???=???1 or 0?<?|??|?<?1), where Q is an n×n positive definite matrix, C is an mn ×mn positive semidefinite matrix, I is an m×m identity matrix, and A is an arbitrary mn×n matrix. This equation is connected with a certain interpolation problem when ???=???1. Using the properties of the Kronecker product and the theory for the monotonic operator defined in a normal cone, we prove the existence and uniqueness of the positive definite solution which is contained in the set {X|I???X?>?C} under the condition that I???Q?>?C. The iterative methods to compute the unique solution is proposed. Numerical examples show that the methods are feasible and effective.     

The product of two dependent random variables with regularly varying or rapidly varying tails

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie-Gumbel-Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of XY. We quantify the impact as the limit, as x→∞, of the quotient of Pr(XY>x) and Pr(X^∗Y^∗>x), where X^∗ and Y^∗ are independent random variables identically distributed as X and Y, respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.     

The Bessel ratio distribution

Let X and Y be two random variables; then the exact distribution of the ratio $X/Y$ is derived when X and Y are independent Bessel function random variables. To cite this article: S. Nadarajah, S. Kotz, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A New Method for Bounding the Distance Between Sums of Independent Integer-Valued Random Variables

Let X ₁, X ₂,..., X _n and Y ₁, Y ₂,..., Y _n be two sequences of independent random variables which take values in ? and have finite second moments. Using a new probabilistic method, upper bounds for the Kolmogorov and total variation distances between the distributions of the sums \(\sum_{i=1}^{n}X_{i}\) and \(\sum_{i=1}^{n}Y_{i}\) are proposed. These bounds adopt a simple closed form when the distributions of the coordinates are compared with respect to the convex order. Moreover, they include a factor which depends on the smoothness of the distribution of the sum of the X _i ’s or Y _i ’s, in that way leading to sharp approximation error estimates, under appropriate conditions for the distribution parameters. Finally, specific examples, concerning approximation bounds for various discrete distributions, are presented for illustration.     

The Equation f(X) = f(Y) in Rational Functions X = X(t), Y = Y(t)

We determine all the complex polynomials f(X) such that, for two suitable distinct, nonconstant rational functions g(t) and h(t), the equality f(g(t)) = f(h(t)) holds. This extends former results of Tverberg, and is a contribution to the more general question of determining the polynomials f(X) over a number field K such that f(X) – has at least two distinct K-rational roots for infinitely many K.     

A generalization of line graphs: (X,Y)-intersection graphs

Given a pair (X, Y) of fixed graphs X and Y, the (X, Y)-intersection graph of a graph G is a graph whose vertices correspond to distinct induced subgraphs of G that are isomorphic to Y, and where two vertices are adjacent iff the intersection of their corresponding subgraphs contains an induced subgraph isomorphic to X. This generalizes the notion of line graphs, since the line graph of G is precisely the (K₁, K₂)-intersection graph of G. In this paper, we consider the forbidden induced subgraph characterization of (X, Y)-intersection graphs for various (X, Y) pairs; such consideration is motivated by the characterization of line graphs through forbidden induced subgraphs. For this purpose, we restrict our attention to hereditary pairs (a pair (X, Y) is hereditary if every induced subgraph of any (X, Y)-intersection graph is also an (X, Y)-intersection graph), since only for such pairs do (X, Y)-intersection graphs have forbidden induced subgraph characterizations. We show that for hereditary 2-pairs (a pair (X, Y) is a 2-pair if Y contains exactly two induced subgraphs isomorphic to X), the family of line graphs of multigraphs and the family of line graphs of bipartite graphs are the maximum and minimum elements, respectively, of the poset on all families of (X, Y)-intersection graphs ordered by set inclusion. We characterize 2-pairs for which the family of (X, Y)-intersection graphs are exactly the family of line graphs or the family of line graphs of multigraphs. © 1996 John Wiley & Sons, Inc.     

Tail asymptotics for dependent subexponential differences

We study the asymptotic behavior of ?(X ? Y > u) as u → ∞, where X is subexponential, Y is positive, and the random variables X and Y may be dependent. We give criteria under which the subtraction of Y does not change the tail behavior of X. It is also studied under which conditions the comonotonic copula represents the worst-case scenario for the asymptotic behavior in the sense of minimizing the tail of X ? Y. Some explicit construction of the worst-case copula is provided in other cases.     

On the product of symmetric random variables

Let X and Y be two independent and nondegenerate random variables such that X is symmetric about a and Y is symmetric about b. It is shown that XY is symmetric about ab if and only if ab= 0.     

Stochastically weighted stochastic dominance concepts with an application in capital budgeting

The problem of comparing random vectors arises in many applications. We propose three new concepts of stochastically weighted dominance for comparing random vectors X and Y. The main idea is to use a random vector V to scalarize X and Y   as V^TX$V^{T}X$ and V^TY$V^{T}Y$, and subsequently use available concepts from stochastic dominance and stochastic optimization for comparison. For the case where the distributions of X, Y and V have finite support, we give (mixed-integer) linear inequalities that can be used for random vector comparison as well as for modeling of optimization problems where one of the random vectors depends on decisions to be optimized. Some advantages of the proposed new concepts are illustrated with the help of a capital budgeting example.     

A Kotz and Steutel type of characterization of the gamma family

Let X and Y be independent identically distributed (i.i.d.) nondegenerate and positive random variables with a common absolutely continuous distribution function F(x). We use the notation Z?=?max(X, Y) and W?=?min(X, Y). In the present paper, we prove that ${\frac{(Z - W)}{(Z + W)}}$ and (Z +?W) are independent if and only if X and Y have gamma distribution.     

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ ^*x exists.     

On the uniform density of C(X) ⊗ C(Y) in C(X × Y)

We prove that if X and Y are compact Hausdorff spaces, then every f ∈ C(X × Y)₊, i.e. f(x, y) ≥ 0 for all (x, y) ∈ X × Y, can be approximated uniformly from below and above by elements of the form , where f_i ∈ C(X)₊ and g_i ∈ C(Y)₊ for i = 1, 2, …, n. The proof uses only elementary topology. We use this result, in conjuction with Kakutani's M-spaces representation theorem, to obtain an alternative proof for a known property of Fremlin's Riesz space tensor product of Archimedean Riesz spaces.