Convex Ordering for Random Vectors using Predictable Representation

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends the results proved in Klein et al. (Electron J Probab 11(20):27, 2006) in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes. The work described in this paper was partially supported by a grant from City University of Hong Kong (Project No. 7200108).     

Ordering Functions of Random Vectors, with Application to Partial Sums

It is known that the sums of the components of two random vectors (X ₁,X ₂,…,X _n ) and (Y ₁,Y ₂,…,Y _n ) ordered in the multivariate (s ₁,s ₂,…,s _n )-increasing convex order are ordered in the univariate (s ₁+s ₂+?+s _n )-increasing convex order. More generally, real-valued functions of (X ₁,X ₂,…,X _n ) and (Y ₁,Y ₂,…,Y _n ) are ordered in the same sense as long as these functions possess some specified non-negative cross-derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S ₁,S ₂,…,S _n ) and (T ₁,T ₂,…,T _n ) where S _j =X ₁+?+X _j and T _j =Y ₁+?+Y _j and we show that these random vectors are ordered in the multivariate (s ₁,s ₁+s ₂,…,s ₁+?+s _n )-increasing convex order. The consequences of these general results for the upper orthant order and the orthant convex order are discussed.     

模糊随机向量凸集的结构和性质

在模糊随机向量凸集概念的基础上,分析了模糊随机向量凸集的结构,研究了模糊随机向量凸集的基本性质.     

Ordering Integer Vectors for Coordinate Deletions

Given a family of sets/vectors of the same cardinality/dimensionyou get the shadow by deleting one element/coordinate from aset/vector in all possible ways. You find the family with thesmallest shadow by ordering all sets/vectors. The set case wassolved by Kruskal (1963), and Katona (1966), and has many applications.We study two orderings which solve the 0, 1 vector case, andgive the shadow size.     

Convex risk functionals: Representation and applications

We introduce the family of law-invariant convex risk functionals, which includes a wide majority of practically used convex risk measures and deviation measures. We obtain a unified representation theorem for this family of functionals. Two related optimization problems are studied. In the first application, we determine worst-case values of a law-invariant convex risk functional when the mean and a higher moment such as the variance of a risk are known. Second, we consider its application in optimal reinsurance design for an insurer. With the help of the representation theorem, we can show the existence and the form of optimal solutions.     

Common Principal Components for Dependent Random Vectors

Let the kp-variate random vector X be partitioned into k subvectors X_i of dimension p each, and let the covariance matrix Ψ of X be partitioned analogously into submatrices Ψ_ij. The common principal component (CPC) model for dependent random vectors assumes the existence of an orthogonal p by p matrix β such that β^tΨ_ijβ is diagonal for all (i, j). After a formal definition of the model, normal theory maximum likelihood estimators are obtained. The asymptotic theory for the estimated orthogonal matrix is derived by a new technique of choosing proper subsets of functionally independent parameters.     

Inequalities for Convex Hulls of Random Points

 We prove an estimate for the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points chosen in a plane convex body. (Received 25 January 2000)     

Inequalities for Convex Hulls of Random Points

 We prove an estimate for the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points chosen in a plane convex body.     

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

Let \(\{X_i, i\ge 1\}\) be i.i.d. \(\mathbb {R}^d\)-valued random vectors attracted to operator semi-stable laws and write \(S_n=\sum _{i=1}^{n}X_i\). This paper investigates precise large deviations for both the partial sums \(S_n\) and the random sums \(S_{N(t)}\), where N(t) is a counting process independent of the sequence \(\{X_i, i\ge 1\}\). In particular, we show for all unit vectors \(\theta \) the asymptotics

$$\begin{aligned} {\mathbb P}(|\langle S_n,\theta \rangle |>x)\sim n{\mathbb P}(|\langle X,\theta \rangle |>x) \end{aligned}$$

which holds uniformly for x-region \([\gamma _n, \infty )\), where \(\langle \cdot , \cdot \rangle \) is the standard inner product on \(\mathbb {R}^d\) and \(\{\gamma _n\}\) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and \(\mathbb {R}^d\)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

     

Sample Covariance Matrix for Random Vectors with Heavy Tails

We compute the asymptotic distribution of the sample covariance matrix for independent and identically distributed random vectors with regularly varying tails. If the tails of the random vectors are sufficiently heavy so that the fourth moments do not exist, then the sample covariance matrix is asymptotically operator stable as a random element of the vector space of symmetric matrices.     

The Central Limit Problem for Random Vectors with Symmetries

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries. This work was done while at Stanford University.     

关于随机紧凸集的更新定理

该文在讨论了多维更新定理的基础上,重点研究了随机紧凸集的Minkowski和的更新定理,得到了一系列重要结论.     

Small-Ball Probabilities for the Volume of Random Convex Sets

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.     

New Representation Formula for Convex Functions on Separable Normed Spaces

We derive a new representation formula for lower-semicontinuous convex functions on separable normed spaces. As a consequence of this formula, we obtain a C -approximation method for convex functions which are not necessarily differentiable.     

Some Comments on Deviation Inequalities for Infinitely Divisible Random Vectors

In this paper, we consider deviation inequalities for infinitely divisible random vectors in R ^k and infinite-dimensional spaces l _p, 1 p We compare the results obtained using the covariance representation for infinitely divisible random vectors with the well-known Talagrand's result on measure concentration phenomenon.     

φ-混合重尾随机向量序列的Chover型重对数律

本文讨论同分布的φ-混合随机向量序列其共同分布属于某个没有Gauss分量的广义的半稳定律的吸引场部分和的积分检验的极限结果,由此可推出相应的Chover型重对数律.     

The Almost Sure Limit Theorem for Sums of Random Vectors

We establish a sufficient condition for an almost sure limit theorem for sums of independent random vectors under minimal moment conditions and assumptions on normalizing sequences. We provide an example showing that our condition is close to the optimal one, as well as a related sufficient condition due to Berkes and Dehling. Bibliography: 5 titles.     

On a Class of Characterization Problems for Random Convex Combinations

We consider stochastic equations of the form X = d W1X + W2X,where (W1, W2), X and X are independent, '=d' denotes equality indistribution, EW1 + EW2 = 1 and X =d X. We discuss existence,uniqueness and stability of the solutions, using contraction arguments andan approach based on moments. The case of {0, 1}-valued W1 and constant W2leads to a characterization of exponential distributions.