On a Characterization Theorem for Locally Compact Abelian Groups Containing an Element of Order 2

According to the well-known Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. While coefficients of linear forms are topological automorphisms of a group.

     

On a characterization theorem for locally compact abelian groups

The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.     

On the Heyde theorem for some locally compact Abelian groups

According to the well-known Heyde theorem Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in n independent random variables given another. For n = 2 we prove analogs of this theorem in the case when random variables take values in a locally compact Abelian group X, and coefficients of the linear forms are topological automorphisms of the group X.     

On a Characterization Theorem on a -adic Solenoids

Doklady Mathematics - According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in independent...     

ON HEYDE’S THEOREM ON THE GROUP $$\mathbb{R}$$ × $$\mathbb{T}$$

Doklady Mathematics - According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of...     

On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008.     

Normal Linear Regression Models With Recursive Graphical Markov Structure

A multivariate normal statistical model defined by the Markov properties determined by an acyclic digraph admits a recursive factorization of its likelihood function (LF) into the product of conditional LFs, each factor having the form of a classical multivariate linear regression model (≡WMANOVA model). Here these models are extended in a natural way to normal linear regression models whose LFs continue to admit such recursive factorizations, from which maximum likelihood estimators and likelihood ratio (LR) test statistics can be derived by classical linear methods. The central distribution of the LR test statistic for testing one such multivariate normal linear regression model against another is derived, and the relation of these regression models to block-recursive normal linear systems is established. It is shown how a collection of nonnested dependent normal linear regression models (≡Wseemingly unrelated regressions) can be combined into a single multivariate normal linear regression model by imposing a parsimonious set of graphical Markov (≡Wconditional independence) restrictions.     

Analytic theory of linear forms of independent random variables

It is shown that in some characterizations of the Gaussian distribution, through natural properties of linear forms of independent, possibly equidistributed random variables, instead of the -algebra generated by one of the forms one can restrict oneself to the finite-dimensional space of polynomials of this form. Thus, the corresponding characterization theorems are significantly refined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 37–44, 1986.     

线性规划在最优火力分配辅助决策中的应用

采用层次分析法和线性规划法对坦克连冲击时目标价值和火力分配问题进行了分析.在理论分析的基础上,用C++Builder开发火力分配辅助决策软件,该软件以报告文件的形式显示目标价值结果,以报告文件或图形的方式显示火力分配结果.     

Use of prior information in the consistent estimation of regression coefficients in measurement error models

A multivariate ultrastructural measurement error model is considered and it is assumed that some prior information is available in the form of exact linear restrictions on regression coefficients. Using the prior information along with the additional knowledge of covariance matrix of measurement errors associated with explanatory vector and reliability matrix, we have proposed three methodologies to construct the consistent estimators which also satisfy the given linear restrictions. Asymptotic distribution of these estimators is derived when measurement errors and random error component are not necessarily normally distributed. Dominance conditions for the superiority of one estimator over the other under the criterion of Löwner ordering are obtained for each case of the additional information. Some conditions are also proposed under which the use of a particular type of information will give a more efficient estimator.     

Application of the method of special series to the representation of solutions of the Lin-Reissner-Tsien equation

For the two-dimensional Lin-Reissner-Tsien equation, which describes nonstationary gas flows, we construct new classes of solutions with functional arbitrariness in the form of series in powers of specially chosen functions. Coefficients of such series are found successively as solutions of linear ordinary differential equations or as solutions of linear partial differential equations. The use of special series whose coefficients are determined by linear partial differential equations allowed us to satisfy two given additional boundary conditions exactly. For one class of flows, these coefficients were found in an explicit form from linear equations of the hyperbolic type; for another one, they were found from linear equations of the parabolic type. This circumstance was used to prove the convergence of such series and to study the asymptotics of the solutions constructed. We present results of numerical calculations on nonstationary transonic flow around a wedge.     

On an optimal solution for maximizing the probability of retention in PPS sampling

A direct alternative proof of Keyfitz's optimal solution to the problem of maximizing the probability of retention in sampling on a second occasion is given, using techniques of elementary linear algebra. The proof and comments help one to better understand Keyfitz's solution, and they clearly demonstrate that the closed form solution of Keyfitz is one of a possible infinity of solutions offered by a linear programming approach. We also give one of those other solutions offered by linear programming, which is easy to obtain by hand calculations using only the operation of subtraction.     

On the Joint Distribution of a Quadratic and a Linear Form in Normal Variables

In this paper a series representation of the joint density and the joint distribution of a quadratic form and a linear form in normal variables is developed. The expansion makes use of Laguerre polynomials. As an example the calculation of the joint distribution of the mean and the sample variance is considered. The truncated series is compared with the empirical distribution function which was determined in a Monte Carlo study.     

Fluid approximations for a processor-sharing queue

In this paper a fluid approximation, also known as a functional strong law of large numbers (FSLLN) for a GI/G/1 queue under a processor-sharing service discipline is established and its properties are analysed. The fluid limit depends on the arrival rate, the service time distribution of the initial customers, and the service time distribution of the arriving customers. This is in contrast to the known result for the GI/G/1 queue under a FIFO service discipline, where the fluid limit is piecewise linear and depends on the service time distribution only through its mean. The piecewise linear form of the limit can be recovered by an equilibrium type choice of the initial service distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on the multidimensional dual

Lexicographic linear goal programming (i.e., the specific form of goal programming wherein one seeks the lexicographic minimum of an ordered set of goal deviations) represents one of the most common and, evidently, widely used forms of goal programming. This form, also described as linear goal programming with a ‘preemptive priority’ structure, has also engendered considerable interest in the literature. However, the dual of the lexicographic linear goal programming model has, thus far, received only minimal exposure. This despite the fact that this dual was developed more than ten years ago. The purpose of this note is to provide a single, unified presentation of the fundamental notion of duality in lexicographic linear goal programming and to indicate existing as well as potential exploitations of this concept.     

Approximate biflow solutions of the kinetic Bryan–Pidduck equation

Some explicit approximate solutions of the non‐linear Bryan–Pidduck equation (that is the Boltzmann equation for the model of rough spheres) are proposed. They have a form of spatially non‐homogeneous linear combination of two global Maxwellians with zero mass angular velocities but arbitrary mass linear velocities. The low‐temperature asymptotics of the uniform‐integral and the pure integral errors between the sides of this equation are found. Sufficient conditions of the infinitesimality of these errors are received, which are based on some requirements on coefficient functions and parameters of the distribution. Copyright © 2000 John Wiley & Sons, Ltd.     

分层广义线性模型在准备金评估中的建模研究

考虑到赔付流量三角形数据同一事故年反复观测的纵向特征以及数据结构的层次性,建立了分层广义线性模型.与通常的随机模型相比,分层广义线性模型不但可以选择条件反应变量的分布而且风险参数分布范围也更加广泛.利用h-似然函数估计分层广义线性模型的模型参数,降低了计算量.为使模型具有可比性,评估模型的预测精度,推导了模型预测误差的估计式.为充分利用已知赔付信息,将赔付额和赔付次数两种赔付信息纳入未决赔款准备金评估模型,建立了两阶段分层广义线性模型.在线性预测量中考虑了各种固定效应和随机效应以及模型结构的散布参数,改进了线性预估量结构.研究表明:分层广义线性模型对于数据的各种分布及形式都具有很好的适应性,更加符合保险实务现实的赔付规律.     

Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Summary. Consider the stationary linear process , , where is an i.i.d. finite variance sequence. The spectral density of may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form , where denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels and for to converge to a Gaussian distribution. We show that this convergence holds if and are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain. Received: 28 February 1996 / In revised form: 10 July 1996     

On the absolute instability of semi-implicit schemes for hydrostatic models

The dependence of the linear stability of two-time-level finite-difference semi-implicit schemes on the choice of reference temperature profile is studied. Particular vertical profiles of the temperature are considered to derive analytical conditions of stability. Analysis is made for general form of different model parameters such as the number of vertical levels and their distribution, the time step size, and the values of the viscosity coefficients. The derived conditions of stability are more restrictive than those for three-time-level schemes, but obtained necessary and sufficient condition for constant vertical lapse rates of the temperature has the form frequently applied to three-time-level schemes: the basic temperature profile should be warmer than the actual one. Performed numerical experiments show that the last restriction is neither necessary nor sufficient condition of stability for general temperature profiles.     

The problem of averaging random structures in terms of distribution functions

A new approach to solving averaging problems for micro-inhomogeneous continua, based on a restatement of the problem in terms of distribution functions, is described. Problems having a variational structure are considered. It is shown that, in terms of distribution functions, they reduce to the problem of minimizing a linear functional, having the meaning of the expectation value of the energy, in a set of distribution functions which is distinguished by an infinite number of linear constants. These constraints express certain matching conditions and contain multipoint distribution functions of the random characteristics of the medium. The constraints form an unlinked chain, the break of which at the n-th step contains only n-point distribution functions. In view of this, a sequence of approximate problems arises.