Surface Measures Generated by Differentiable Measures

We study surface measures on level sets of functions on general probability spaces with measures differentiable along vector fields and suggest a new simple construction. Our construction applies also to level sets of mappings with values in finite-dimensional spaces. The standard surface measures arising for Gaussian measures in the Malliavin calculus can be obtained in this way. A positive answer is given to a question raised by M. Röckner concerning continuity of surface measures with respect to a parameter.     

Absolute Curvature Measures

For arbitrary sets of positive reach in euclidean space a new kind of absolute curvature measures is introduced. These measures possess similar section and projection properties as their signed counterparts—the Lipschitz-Killing curvature measures. In the present paper interpretations as invariant measures of the sets of colliding planes and as mean projection measures are given.     

Informational Relationship Measures

This paper gives the definitions of ten normed information ratesas measures of relationship between two random variables. Thebehaviour of these measures have been considered in the caseof a bivariate uniform model. Numerical comparisons betweenthe measures are made, in order to choose a measure which hassome advantages over the other measures.     

Hessian Measures of Convex Functions and Applications to Area Measures

The Hessian measures of a (semi-)convex function can be introducedas coefficients of a local Steiner formula. The investigationof Hessian measures is continued by the provision of a geometriccharacterization of the support of these measures. Then theRadon–Nikodym derivative and the absolute continuity ofHessian measures with respect to Lebesgue measure are explored.As special cases of the results, known results for surface areameasures of convex bodies are recovered.     

On Coherent Risk Measures Induced by Convex Risk Measures

We study the close relationship between coherent risk measures and convex risk measures. Inspired by the obtained results, we propose a class of coherent risk measures induced by convex risk measures. The robust representation and minimization problem of the induced coherent risk measure are investigated. A new coherent risk measure, the Entropic Conditional Value-at-Risk (ECVaR), is proposed as a special case. We show how to apply the induced coherent risk measure to realistic portfolio selection problems. Finally, by comparing its out-of-sample performance with that of CVaR, entropic risk measure, as well as entropic value-at-risk, we carry out a series of empirical tests to demonstrate the practicality and superiority of the ECVaR measure in optimal portfolio selection.     

Measures of Asymmetry Dual to Mean Minkowski Measures of Asymmetry for Convex Bo dies

We introduce a family of measures (functions) of asymmetry for convex bodies and discuss their properties. It turns out that this family of measures shares many nice properties with the mean Minkowski measures. As the mean Minkowski measures describe the symmetry of lower dimensional sections of a convex body, these new measures describe the symmetry of lower dimensional orthogonal projections.     

Lacunary Measures and Self-similar Probability Measuresin Function Spaces

The paper deals with multifractal quantities for some types of Radon measures, especially self-similar probability measures, and their relations to Besov spaces.     

Super-Tree Random Measures

We use supercritical branching processes with random walk steps of geometrically decreasing size to construct random measures. Special cases of our construction give close relatives of the super-(spherically symmetric stable) processes. However, other cases can produce measures with very smooth densities in any dimension.     

Hessian Measures III

In this paper, we continue previous investigations into the theory of Hessian measures. We extend our weak continuity result to the case of mixed k-Hessian measures associated with k-tuples of k-convex functions, on domains in Euclidean n-space, k=1,2,…,n. Applications are given to capacity, quasicontinuity, and the Dirichlet problem, with inhomogeneous terms, continuous with respect to capacity or combinations of Dirac measures.     

Systematic Measures of Biological Networks I: Invariant Measures and Entropy

This paper is part I of a two‐part series devoted to the study of systematic measures in a complex biological network modeled by a system of ordinary differential equations. As the mathematical complement to our previous work with collaborators, the series aims at establishing a mathematical foundation for characterizing three important systematic measures: degeneracy, complexity, and robustness, in such a biological network and studying connections among them. To do so, we consider in part I stationary measures of a Fokker‐Planck equation generated from small white noise perturbations of a dissipative system of ordinary differential equations. Some estimations of concentration of stationary measures of the Fokker‐Planck equation in the vicinity of the global attractor are presented. The relationship between the differential entropy of stationary measures and the dimension of the global attractor is also given.© 2016 Wiley Periodicals, Inc.     

Weak Convergence of Topological Measures

Journal of Theoretical Probability - Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and...     

Tail Properties of Correlation Measures

We study the tail properties of a class of Borel probability measures, called correlation measures. We show that (i) there exist correlation measures with exponentially decaying tail probabilities, and (ii) roughly speaking, no correlation measure may have smaller tail probabilities than a Gaussian measure.     

Stationary Random Measures on Homogeneous Spaces

This paper discusses stationary random measures on a homogeneous space and their Palm measures. It starts with such fundamental properties as the refined Campbell theorem and then proceeds to consider invariant transports, invariance and transport properties of Palm measures, and stationary partitions. A key tool is a transformation of random measures that permits the extension of recent results for stationary random measures on a group to the more general case of stationary random measures on a homogeneous state space.     

Banach空间上集值测度与模糊数测度的扩张

在Banach空间上,给出集值测度的扩张定理并借助集测度的扩张给出了模糊数测度的扩张定理。     

Measures on infinite dimensional Grassmann manifolds

A 1-parameter family of quasi-invariant measures is presented. These measures are cylinder measures in graph coordinates. Their characteristic functions are represented as integrals relative to an infinite product measure. This is applied to the problem of determining the support properties of the measures. One of the measures can be used to define the unitary structure for the basic representation of the affine extension of the restricted unitary group.     

Measures on the Random Graph

We consider the problem of characterizing the finitely additiveprobability measures on the definable subsets of the randomgraph which are invariant under the action of the automorphismgroup of this graph. We show that such measures are all integralsof Bernoulli measures (which arise from the coin-flipping modelof the construction of the random graph). We also discuss generalizationsto other theories.     

Measures on Graphs and Groupoid Measures

The main purpose of this paper is to introduce several measures determined by a given finite directed graph. To construct σ-algebras for those measures, we consider several algebraic structures induced by G; (i) the free semigroupoid of the shadowed graph (ii) the graph groupoid of G, (iii) the disgram set and (iv) the reduced diagram set . The graph measures determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures determined by (iii) is the diagram measure measuring how long we moved consequently from the starting positions (which are vertices) of some movements on G. The graph measures and determined by (ii) and (iv) are the (graph) groupoid measure and the (quotient-)groupoid measure, respectively. We show that above graph measurings are invariants on shadowed graphs of finite directed graphs. Also, we will consider the reduced diagram measure theory on graphs. In the final chapter, we will show that if two finite directed graphs G ₁ and G ₂ are graph-isomorphic, then the von Neumann algebras L ^∞(μ ₁) and L ^∞(μ ₂) are *-isomorphic, where μ ₁ and μ ₂ are the same kind of our graph measures of G ₁ and G ₂, respectively. Received: December 7, 2006. Revised: August 3, 2007. Accepted: August 18, 2007.     

Measures on finite concrete logics

We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets.

     

Orlicz–Young Structures of Young Measures

The paper examines the integration of Young functions applied to Young measures and identifies Orlicz-like structures in the space of Young measures. In particular, a convergence intermediate between the weak convergence of measures and the variational norm is determined; it serves in the completion of the Orlicz space of functions when interpreted as degenerate Young measures. Partial linear operations are defined on Young measures with respect to which the linear operations in the Orlicz space of functions are continuously embedded in the space of Young measures. This leads to a definition of convexity-type structures in the space of Young measures via a limiting procedure. These structures enable applications of Young functions arguments to Young measures. Applications to optimal control and to well posedness of minimization in function spaces with respect to convex functions are provided.     

Products of Capacities Derived from Additive Measures

One of the unanswered questions in non-additive measure theory is how to define product of non-additive measures. Most of the approaches that have already been presented only work for discrete measures. In this paper a new approach is presented for not necessarily discrete non-additive measures that are in a certain relation with additive measures, usually this means that they are somehow derived from the additive measures.