On rank correlation measures for non-continuous random variables

For continuous random variables, many dependence concepts and measures of association can be expressed in terms of the corresponding copula only and are thus independent of the marginal distributions. This interrelationship generally fails as soon as there are discontinuities in the marginal distribution functions. In this paper, we consider an alternative transformation of an arbitrary random variable to a uniformly distributed one. Using this technique, the class of all possible copulas in the general case is investigated. In particular, we show that one of its members—the standard extension copula introduced by Schweizer and Sklar—captures the dependence structures in an analogous way the unique copula does in the continuous case. Furthermore, we consider measures of concordance between arbitrary random variables and obtain generalizations of Kendall's tau and Spearman's rho that correspond to the sample version of these quantities for empirical distributions.     

Cycle-transitive comparison of independent random variables

The discrete dice model, previously introduced by the present authors, essentially amounts to the pairwise comparison of a collection of independent discrete random variables that are uniformly distributed on finite integer multisets. This pairwise comparison results in a probabilistic relation that exhibits a particular type of transitivity, called dice-transitivity. In this paper, the discrete dice model is generalized with the purpose of pairwisely comparing independent discrete or continuous random variables with arbitrary probability distributions. It is shown that the probabilistic relation generated by a collection of arbitrary independent random variables is still dice-transitive. Interestingly, this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Furthermore, when the marginal distributions of the random variables belong to the same parametric family of distributions, the probabilistic relation exhibits interesting types of isostochastic transitivity, such as multiplicative transitivity. Finally, the probabilistic relation generated by a collection of independent normal random variables is proven to be moderately stochastic transitive.     

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.     

关于任意随机变量序列泛函的一类强偏差定理

通过概率空间上的任意随机变量的分布与独立分布的比较.研究任意随机变量序列泛函的强偏差定理,即小偏差定理.将已有的某些连续型及离散随机变量序列的强偏差定理加以推广.     

Tail dependence for elliptically contoured distributions

The relationship between the theory of elliptically contoured distributions and the concept of tail dependence is investigated. We show that bivariate elliptical distributions possess the so-called tail dependence property if the tail of their generating random variable is regularly varying, and we give a necessary condition for tail dependence which is somewhat weaker than regular variation of the latter tail. In addition, we discuss the tail dependence property for some well-known examples of elliptical distributions, such as the multivariate normal, t, logistic, and Bessel distributions.     

关于任意非负随机序列随机和的一类随机偏差定理

本文引入任意随机变量序列随机极限对数似然比概念,作为任意相依随机序列联合分布与其边缘乘积分布“不相似”性的一种度量,利用构造新的密度函数方法来建立几乎处处收敛的上鞅,在适当的条件下,给出了任意受控随机序列的一类随机偏差定理.     

On Bernoulli decompositions for random variables, concentration bounds, and spectral localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: (i) an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and (ii) a proof, based on the Bernoulli case, of spectral localization for random Schrödinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.     

An order-statistics-based method for constructing multivariate distributions with fixed marginals

A new system of multivariate distributions with fixed marginal distributions is introduced via the consideration of random variates that are randomly chosen pairs of order statistics of the marginal distributions. The distributions allow arbitrary positive or negative Pearson correlations between pairs of random variates and generalise the Farlie–Gumbel–Morgenstern distribution. It is shown that the copulas of these distributions are special cases of the Bernstein copula. Generation of random numbers from the distributions is described, and formulas for the Kendall and grade (Spearman) correlations are given. Procedures for data fitting are described and illustrated with examples.     

关于Shannon—Mcmillan定理的若干研究（Ⅱ）

该文通过概率空间上的任意分布列与独立分布列比较，研究任意随机变量序列相对熵密度用不等式给出的强极限定理，即小偏差定理，并由此得出若干Shannon－Mcmillan定理，将作者已有的关于离散信源的结果加以推广.     

关于二项分布的强偏差定理

利用分析方法建立了用不等式表示的用渐近平均对数似然比刻划的服从二项分布的随机变量序列的强偏差定理,作为推论得到了服从二项分布的相依随机变量序列的强大数定律.     

Central Regions and Dependency

The paper introduces an approach to the ordering of dependence which is based on central regions. A d-variate probability distribution is described by a nested family of sets, called central regions. Those regions are affine equivariant, compact and starshaped and concentrate about a properly defined center. They can be seen as level sets of a depth function. Special cases are Mahalanobis, zonoid, and likelihood regions. A d-variate distribution is called more dependent than another one if the volume of each central region is smaller with the first distribution. This dependence order is characterized by an inequality between determinants of certain parameter matrices if either (i) F and G are arbitrary distributions and the central regions are Mahalanobis or (ii) F and G belong to an elliptical family of distributions and the central regions are arbitrary. If the regions are zonoid regions, the dependence order implies the ordering of lift zonoid volumes. Alternatively, the dependence order is applied to the copulae of the given distributions. Generalized correlation indices are proposed which are increasing with the dependence orders.     

Quenched Free Energy and Large Deviations for Random Walks in Random Potentials

We study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps. The potential can be unbounded and can depend on a few steps of the walk. Directed, undirected, and stretched polymers, as well as random walk in random environment, are covered. The restriction needed is on the moment of the potential, in relation to the degree of mixing of the ergodic environment. We derive two variational formulas for the limiting quenched free energy and prove a process‐level quenched large deviation principle (LDP) for the empirical measure. As a corollary we obtain LDPs for types of random walks in random environments not covered by earlier results. © 2012 Wiley Periodicals, Inc.     

Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables

In this paper we study the asymptotic behavior of the tail probabilities of sums of dependent and real-valued random variables whose distributions are assumed to be subexponential and not necessarily of dominated variation. We propose two general dependence assumptions under which the asymptotic behavior of the tail probabilities of the sums is the same as that in the independent case. In particular, the two dependence assumptions are satisfied by multivariate Farlie-Gumbel-Morgenstern distributions.     

On closure properties of heavy-tailed distributions for random sums*

We investigate some closure properties of heavy-tailed distributions for random sums. We first consider the random sums with independent or some certain dependent long-tailed increments. We also consider the real-valued increments with dominatedly varying-tailed distributions under extended negative dependence and obtain two results on necessary and sufficient conditions or sufficient conditions for the closure on random sums.     

Duration dependence models for claim counts

The aim of the paper is to introduce new claim count distributions constructed from different waiting time assumptions, such as the Exponential, Gamma and Weibull distributions. These models are then fitted to panel data with Gamma distributed random effects. The random effects allow for serial dependence and take residual heterogeneity into account. Predictive distributions are obtained with the help of Markov Chain Monte Carlo simulations. The approach is illustrated on the basis of a Belgian motor third party liability insurance portfolio observed for three years.     

Second order risk aggregation with the Bernstein copula

We analyze the tail of the sum of two random variables when the dependence structure is driven by the Bernstein family of copulas. We consider exponential and Pareto distributions as marginals. We show that the first term in the asymptotic behavior of the sum is not driven by the dependence structure when a Pareto random variable is involved. Consequences on the Value-at-Risk are derived and examples are discussed.     

Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients

The mean and variance of the temperature are analytically obtained in a functionally graded annular disc with spatially random heat transfer coefficients (HTCs) on the upper and lower surfaces. This annular disc has arbitrary variations in the HTCs (i.e., arbitrary thermal interaction with the surroundings) and gradient material composition only along the radial direction and is subjected to deterministic axisymmetrical heating at the lateral surfaces. The stochastic temperature field is analysed by considering the annular disc to be multilayered with spatially constant material properties and spatially constant but random HTCs in each layer. A type of integral transform method and a perturbation method are employed in order to obtain the analytical solutions for the statistics. The correlation coefficients of the random HTCs are expressed in the form of a linear function with respect to the radial distance as a non-homogeneous random field of discrete space. Numerical calculations are performed for functionally graded annular discs composed of stainless steel and ceramic, which comprise two types of material composition distributions. The effects of the magnitude of the means of HTCs, volume fraction distributions of the constitutive materials and correlation strengths of the HTCs on the standard deviation of the temperature are discussed.     

Tail dependence between order statistics

In this work, we introduce the s,k-extremal coefficients for studying the tail dependence between the s-th lower and k-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient η of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ?, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.     

Multivariate Discrete Distributions with a Product-Type Dependence

A discrete multivariate probability distribution for dependent random variables, which contains the Poisson and Geometric conditionals distributions as particular cases, is characterized by means of conditional expectations of arbitrary one-to-one functions. Independence of the random variables is also characterized in terms of these conditional expectations. For certain exchangeable and partially exchangeable random variables with a joint distribution of this form it is shown that maximum likelihood estimates coincide with the simple method of moments estimates, suggesting that these models offer a pragmatic way to analyze certain dependent data.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.