离散寿命分布类的统计封闭性

由于几何分布的无记忆性,使得几何分布在离散型寿命分布研究中起着极其重要的作用.文章给出几何分布的一个重要特性———其第一个次序统计量仍服从几何分布(称其满足统计封闭特性).针对不同的离散型寿命分布类研究其是否具有统计封闭性.     

负二项分布类的条件概率封闭性

在研究只允许部分服务台进入休假状态的多服务台M／M／c排队系统时，我们发现了条件Erlang分布的一些有趣的性质，进一步研究我们发现相对应离散随机状态的负二项分布也具有很好的性质（概率封闭性．本文证明了一类负二项分布的概率封闭性．它们对导出复杂排队系统中离散状态下顾客等待时问分布及保险公司中破产概率上界的计算起着重要作用．     

A Skew Laplace Distribution on Integers

We propose a discrete version of the skew Laplace distribution. In contrast with the discrete normal distribution, here closed form expressions are available for the probability density function, the distribution function, the characteristic function, the mean, and the variance. We show that this distribution on integers shares many properties of the skew Laplace distribution on the real line, including unimodality, infinite divisibility, closure properties with respect to geometric compounding, and a maximum entropy property. We also discuss statistical issues of estimation under this model.     

On generalized Stieltjes-Wigert and related orthogonal polynomials

The Charlier, Wall, and generalized Stieltjes-Wigert polynomials are characterized by a property involving the concept of kernel polynomials. This characterization leads to consideration of a certain functional equation satisfied by solutions of the associated Stieltjes moment problem. All distribution functions which satisfy this functional equation are found up to singular functions. This yields new distribution functions, both discrete and absolutely continuous, with respect to which generalized Stieltjes-Wigert polynomials are orthogonal.     

A Note on a Traffic Counting Distribution

In this paper the author formulates and derives a discrete traffic counting distribution and a passing rule which lead to bunches of zero or one “fast” vehicle behind a “slow” vehicle. “Slow” vehicles are distributed in a Poisson fashion and, except for those instants at which passing occurs, the constrained or “fast” vehicles follow immediately behind a “slow” one. The counting distribution is a simple example of one in which successive inter-vehicle spacings have the Markov property.     

二维随机变量条件分布函数教学的思考

本文利用条件概率的定义,由随机变量分布函数的性质,给出一般情形下随机变量条件分布函数的定义,以帮助学生更好地理解随机变量的条件分布函数的概念.     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

Discrete-time GI/Geo/1 queue with multiple working vacations

Consider the discrete time GI/Geo/1 queue with working vacations under EAS and LAS schemes. The server takes the original work at the lower rate rather than completely stopping during the vacation period. Using the matrix-geometric solution method, we obtain the steady-state distribution of the number of customers in the system and present the stochastic decomposition property of the queue length. Furthermore, we find and verify the closed property of conditional probability for negative binomial distributions. Using such property, we obtain the specific expression for the steady-state distribution of the waiting time and explain its two conditional stochastic decomposition structures. Finally, two special models are presented.      

Lower semicontinuity property of multiparameter optimal stopping value and its application to multiparameter prophet inequalities

This paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. The optimal stopping value of a discrete time multiparameter integrable stochastic process whose negative part is uniformly integrable, is lower semicontinuous for the topology of convergence in distribution. The multiparameter version of prophet inequality for the one-parameter optimal stopping problem is formulated and the lower semicontinuity property of the optimal stopping value is applied to the multiparameter prophet inequality.     

Discrete Compound Poisson Process with Curved Boundaries: Polynomial Structures and Recursions

This paper provides a review of recent results, most of them published jointly with Ph. Picard, on the exact distribution of the first crossing of a Poisson or discrete compound Poisson process through a given nondecreasing boundary, of curved or linear shape. The key point consists in using an underlying polynomial structure to describe the distribution, the polynomials being of generalized Appell type for an upper boundary and of generalized Abel–Gontcharoff type for a lower boundary. That property allows us to obtain simple and efficient recursions for the numerical determination of the distribution.      

Preserving Positive Realness Through Discretization

The importance of positive real transfer functions relies on the fact that they are associated with positive linear systems. Those systems possess the property that their input-output product time-integral, which is a measure of the total enerty, is nonnegative. Such a property can be also formulated in the discrete context. It is shown that a discrete positive real transfer function is obtained from a positive real continuous one of relative order zero being strictly stable poles via discretization by a sampler and zero-order hold device provided that the direct input-output transmission gain is sufficiently large. It is also proved that a discrete positive real transfer function may be obtained from a stable continuous one of relative order zero and high direct input-output gain which posses simple complex conjugate critically stable poles even in the case that this one is not positive real. For that purpose, the use of an appropriate phase-lag or phase lead compensating network for the continuous transfer function may be required to ensure positive realness of the discrete transfer function.     

A STRONG ERGODIC THEOREM FOR SOME NONLINEAR MATRIX MODELS FOR THE DYNAMICS OF STRUCTURED POPULATIONS

A general class of matrix difference equation models for the dynamics of discrete class structured populations in discrete time which possess a certain general type of nonlinearity introduced by Leslie for age-structured populations is considered. Arbitrary structuring is allowed in that transitions between any two classes are permitted. It is shown that normalized class distributions for such nonlinear models globally approach a “stable class distribution” and thus possess a strong ergodic property exactly like that of the classical linear theory of demography. However, unlike in the linear theory according to which the total population size grows or dies exponentially, the dynamics of total population size in these nonlinear models are shown to be governed by a nonlinear, nonautonomous scalar difference equation. This difference equation is asymptotically autonomous, and theorems which relate the dynamics of total population size to those of this limiting equation are proved. Examples in which the results are applied to some nonlinear age-structure models found in the literature are given.     

Approximation properties of fine hyperbolic graphs

In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa’s to prove that a fine hyperbolic graph has the metric invariant translation approximation property.     

A Property of the mean deviation (Of some discrete distributions)

Summary This note describes an interesting property of the mean deviation which holds for a number of commonly known discrete distributions. The property is also examined for some of the well-known continuous distributions.     

Property A for groups acting on metric spaces

Gouliang Yu has introduced a property of discrete metric spaces and groups called property A which implies the coarse Baum–Connes Conjecture and hence the Novikov Higher Signature Conjecture. In this paper we extend a result of Jean-Louis Tu to conclude that a group acting by isometries on a metric space with finite asymptotic dimension whose d-stabilizers have property A, also has property A. As a result, we conclude a theorem of Tu, according to which, a fundamental group of a finite graph of groups whose vertices have property A also has property A.     

A discrete Hartley transform based on Simpson's rule

The Hartley transform is an integral transformation that maps a real valued function into a real valued frequency function via the Hartley kernel, thereby avoiding complex arithmetic as opposed to the Fourier transform. Approximation of the Hartley integral by the trapezoidal quadrature results in the discrete Hartley transform, which has proven a contender to the discrete Fourier transform because of its involutory nature. In this paper, a discrete transform is proposed as a real transform with a convolution property and is an alternative to the discrete Hartley transform. Copyright © 2015 John Wiley & Sons, Ltd.     

Random invariant measures for Markov chains,and independent particle systems

Summary Let P be the transition operator for a discrete time Markov chain on a space S. The object of the paper is to study the class of random measures on S which have the property that MP=M in distribution. These will be called random invariant measures for P. In particular, it is shown that MP=M in distribution implies MP=M a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to P. Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.Research supported in part by N.S.F. Grant No. MCS 77-02121     

The lower neighborhoods problem in a space with closure and a finitary theorem

In this paper we study conditions under which all finitely generated subsets in a space (i.e., in a set with the closure property) have finite lower neighborhoods. We prove the finitary property of such a space. In addition, we state the Birchoff and Frink finatary theoremmore precisely. Considered questions have applications to discrete functional systems with superpositions.     

Weighted Poisson Distributions for Overdispersion and Underdispersion Situations

The main goal of this paper is to introduce new exponential families, that come from the concept of weighted distribution, that include and generalize the Poisson distribution. In these families there are distributions with index of dispersion greater than, equal to or smaller than one. This property makes them suitable to fit discrete data in overdispersion or underdispersion situations. We study the statistical properties of the families and we provide a useful interpretation of the parameters. Two classical examples are considered in order to compare the fits with some other distributions. To obtain the fits with the new family, the study of the profile log-likelihood is required.