Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at     

Change point estimation in non-monotonic aging models

The problem of estimating change points in various non-monotonic aging models is considered. A general methodology for consistent estimation of the change point is developed and applied to non-monotonic aging models based on the hazard rate function as well as on the mean residual life function.     

On lower bounds for theL 2-Wasserstein metric in a Hilbert space

We provide two families of lower bounds for theL ²-Wasserstein metric in separable Hilbert spaces which depend on the basis chosen for the space. Then we focus on one of these families and we provide a necessary and sufficient condition for the supremum in it to be attained. In the finite dimensional case, we identify the basis which provides the most accurate lower bound in the family.Research partially supported by the Spanish DGICYT under grants PB91-0306-02-00, 01 and 02.     

Discrete semi-stable distributions

The purpose of this paper is to introduce and study the concepts of discrete semi-stability and geometric semi-stability for distributions with support inZ ₊. We offer several properties, including characterizations, of discrete semi-stable distributions. We establish that these distributions posses the property of infinite divisibility and that their probability generating functions admit canonical representations that are analogous to those of their continuous counterparts. Properties of discrete geometric semi-stable distributions are deduced from the results obtained for discrete semi-stability. Several limit theorems are established and some examples are constructed.     

用聚苯乙烯小球模拟生物组织中的光强分布

用水中周期排列的直径为3μm的单层聚苯乙烯小球以及单个聚苯乙烯小球模拟对光敏感的生物组织，如绿色植物叶肉细胞、动物皮肤的表皮细胞、苍蝇蜜蜂等的复眼以及一种海蛇尾背部的光敏器官的光强分布.通过观测聚苯乙烯小球在不同深度的光强分布，发现随着深度的改变其光强分布是截然不同的.在小球的横截面上，光强分布不很均匀而且呈有规律的花纹图案.在小球的外面，光强分布表现为独立的透镜成像分布.周期排列的小球与单个小球在其横截面上的光强分布也完全不同，前者的光强分布表现出球与球之间很大的相关性. 关键词： 生物组织 周期排列 光强分布     

Explicit ruin formulas for models with dependence among risks

We show that a simple mixing idea allows one to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.     

Optimum selection of the dental implant diameter and length in the posterior mandible with poor bone quality – A 3D finite element analysis

This study aimed to evaluate continuous and simultaneous variations of dental implant diameter and length, and to identify their relatively optimal ranges in the posterior mandible under biomechanical consideration. A 3D finite element model of a posterior mandibular segment with dental implant was created. Implant diameter ranged from 3.0 to 5.0 mm, and implant length ranged from 6.0 to 16.0 mm. The results showed that under axial load, the maximum Von Mises stresses in cortical and cancellous bones decreased by 76.53% and 72.93% respectively, with the increasing of implant diameter and length; and under buccolingual load, by 83.97% and 84.93%, respectively. Under both loads, the maximum displacements of implant-abutment complex decreased by 58.09% and 75.53%, respectively. The results indicate that in the posterior mandible, implant diameter plays more significant roles than length in reducing cortical bone stress and enhancing implant stability under both loads. Meanwhile, implant length is more effective than diameter in reducing cancellous bone stress under both loads. Moreover, biomechanically, implant diameter exceeding 4.0 mm and implant length exceeding 12.0 mm is a relatively optimal combination for a screwed implant in the posterior mandible with poor bone quality.     

Zero Expectile Processes and Bayesian Spatial Regression

We introduce new classes of stationary spatial processes with asymmetric, sub-Gaussian marginal distributions using the idea of expectiles. We derive theoretical properties of the proposed processes. Moreover, we use the proposed spatial processes to formulate a spatial regression model for point-referenced data where the spatially correlated errors have skewed marginal distribution. We introduce a Bayesian computational procedure for model fitting and inference for this class of spatial regression models. We compare the performance of the proposed method with the traditional Gaussian process-based spatial regression through simulation studies and by applying it to a dataset on air pollution in California.     

Alternative Entropy Measures and Generalized Khinchin–Shannon Inequalities

The Khinchin–Shannon generalized inequalities for entropy measures in Information Theory, are a paradigm which can be used to test the Synergy of the distributions of probabilities of occurrence in physical systems. The rich algebraic structure associated with the introduction of escort probabilities seems to be essential for deriving these inequalities for the two-parameter Sharma–Mittal set of entropy measures. We also emphasize the derivation of these inequalities for the special cases of one-parameter Havrda–Charvat’s, Rényi’s and Landsberg–Vedral’s entropy measures.     

The distribution of nodes of given degree in random trees

Let 𝒯_n denote the set of unrooted unlabeled trees of size n and let k ≥ 1 be given. By assuming that every tree of 𝒯_n is equally likely, it is shown that the limiting distribution of the number of nodes of degree k is normal with mean value ∼ μ_kn and variance ∼ σn with positive constants μ_k and σ_k. Besides, the asymptotic behavior of μ_k and σ_k for k → ∞ as well as the corresponding multivariate distributions are derived. Furthermore, similar results can be proved for plane trees, for labeled trees, and for forests. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 227–253, 1999