长尾宽上限相依的不同分布的随机变量和的精确大偏差

研究了服从长尾分布族上的随机变量和的精确大偏差问题,其中假设代表索赔额的随机变量序列是一列宽上限相依的、不同分布的随机变量序列。在给定一些假设条件下,得到了部分和与随机和的两种一致渐近结论。     

New examples of heavy-tailed O-subexponential distributions and related closure properties

Let L and S denote the classes of distributions with long tails and subexponential tails respectively. Let OS denote the class of distributions with O-subexponential tails, which means the distributions with the tails having the same order as the tails of their 2-fold convolutions. In this paper, we first construct a family of distributions without finite means in L∩OS?S. Next some distributions in L∩OS?S, which possess finite means or even finite higher moments, are also constructed. In connection with this, we prove that the class OS is closed under minimization of random variables. However, it is not closed under maximization of random variables.     

Exponential Mixture Representation of Geometric Stable Distributions

We show that every strictly geometric stable (GS) random variable can be represented as a product of an exponentially distributed random variable and an independent random variable with an explicit density and distribution function. An immediate application of the representation is a straightforward simulation method of GS random variables. Our result generalizes previous representations for the special cases of Mittag-Leffler and symmetric Linnik distributions.     

The problem of identification of parameters by the distribution of the maximum random variable

Suppose that X₁, X₂,…, X_n are independently distributed according to certain distributions. Does the distribution of the maximum of {X₁, X₂,…, X_n} uniquely determine their distributions? In the univariate case, a general theorem covering the case of Cauchy random variables is given here. Also given is an affirmative answer to the above question for general bivariate normal random variables with non-zero correlations. Bivariate normal random variables with nonnegative correlations were considered earlier in this context by T. W. Anderson and S. G. Ghurye.     

无限级Dirichlet级数

本文研究了右半平面上无限级的Ｄｉｒｉｃｈｌｅｔ级数及随机Ｄｉｒｉｃｈｌｅｔ级数．这里我们给出一个较宽的系数条件,并证明在一定意义上是最好的;计算无限级Ｄｉｒｉｃｈｌｅｔ级数的精确级;把随机级数的研究引向一般得多的非同分布情况,并得到右半平面上非同分布的无限级随机Ｄｉｒｉｃｈｌｅｔ级数几乎必然（ａ．ｓ．）以虚轴上的每一点为没有有限例外值的Ｂｏｒｅｌ点的结论．     

Rational reconstruction of frailty-based mortality models by a generalisation of Gompertz’ law of mortality

A generalisation of Gompertz’ distribution is proposed, and it is shown that continuous heterogeneous mortality models with Gamma distributed frailty have lifetime random variables distributed as the difference of two such generalised Gompertz random variables. With this result, limitations of existing frailty-based mortality models are identified. The approach taken in this paper allows the frailty distribution to be interpreted as a lifetime reduction distribution and enables application of heterogeneous survival models with a stronger relation to empirically identifiable concepts.     

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.     

A note on runs of geometrically distributed random variables

Recently, Grabner et al. [Combinatorics of geometrically distributed random variables: run statistics, Theoret. Comput. Sci. 297 (2003) 261-270] and Louchard and Prodinger [Ascending runs of sequences of geometrically distributed random variables: a probabilistic analysis, Theoret. Comput. Sci. 304 (2003) 59-86] considered the run statistics of geometrically distributed independent random variables. They investigated the asymptotic properties of the number of runs and the longest run using the corresponding probability generating functions and a Markov chain approach. In this note, we reconsider the asymptotic properties of such statistics using another approach. Our approach of finding the asymptotic distributions is based on the construction of runs in a sequence of m-dependent random variables. This approach enables us to find the asymptotic distributions of many run statistics via the theorems established for m-dependent sequence of random variables. We also provide the asymptotic distribution of the total number of non-decreasing runs and the longest non-decreasing run.     

A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications

We obtain an explicit representation for joint distribution of two-valued random variables with given marginals and for a copula corresponding to such random variables. The results are applied to prove a characterization of r-independent two-valued random variables in terms of their mixed first moments. The characterization is used to obtain an exact estimate for the number of almost independent random variables that can be defined on a discrete probability space and necessary conditions for a sequence of r-independent random variables to be stationary.     

依概率收敛与依分布收敛的关系

本文探讨了随机变量序列依概率收敛与依分布收敛的关系 ,并给出了一个依分布收敛能保证依概率收敛的最弱的条件 ,即 :设分布函数列 { Fn(x) }弱收敛于连续的分布函数 F(x) ,则存在随机变量序列{ξn}和随机变量ξ,它们分别以 { Fn(x) }和 F(x)为其对应的分布函数列和分布函数 ,且 {ξn}依概率收敛于ξ.     

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR SUMS OF NEGATIVELY DEPENDENT RANDOM VARIABLES

In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (-∞, +∞). The results show that for some non-identical random variables, precise large deviations and moderate deviations remain insensitive to negative dependence structure.     

An alternative method for computing mean and covariance matrix of some multivariate distributions

Computing the mean and covariance matrix of some multivariate distributions, in particular, multivariate normal distribution and Wishart distribution are considered in this article. It involves a matrix transformation of the normal random vector into a random vector whose components are independent normal random variables, and then integrating univariate integrals for computing the mean and covariance matrix of a multivariate normal distribution. Moment generating function technique is used for computing the mean and covariances between the elements of a Wishart matrix. In this article, an alternative method that uses matrix differentiation and differentiation of the determinant of a matrix is presented. This method does not involve any integration.     

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.     

基于对称随机变量构造的反例

应用相关文献中对称随机变量分布函数的充要条件,阐明连续型对称随机变量概率密度的偶函数特点,以及对称随机变量的不相关性,构造一些教学反例.     

整值随机变量序列的一类强律

本文引进似然比作为整值随机变量序列相对于服从Poisson分布的独立随机变量序列的偏差的一种度量，并通过限制似然比给出了样水空间的某种子集．在这种子集上得到了一类用不等式表示的强律，独立随机变量序列的一类强律是其特例．     

有限域上随机变量联合分布及二阶矩的分解与应用

本文给出了有限域上随机变量联合概率和二阶矩的分解公式，给出了有限域上随机变量相互独立的谱刻划，应用上述结果，建立了在进行频次分析时，对有限域上随机向量构造的Χ平方统计量与该随机向量坐标函数的非零线性组合的Χ平方统计量之间的内在联系，给出了有限域上相关免疫函数谱特征的新证明，建立了有限域上多输出函数的差分分布与其广义Chrestenson循环谱之间的内在联系，建立了多输出函数的平衡性其差分分布之间的内在联系。     

一类非离散非连续的随机变量的概率分布

本文定义了一类非离散非连续的随机变量的概率分布,使其概率分布与离散型和连续型随机变量的概率分布表示保持一致,并举例求出考研中涉及过的该类型的随机变量的概率分布.     

An algebra of Stein operators

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are kth degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of k independent symmetric variance-gamma distributed random variables.     

封闭型保单组未来损失现值向量的联合分布及渐近分布

本文针对封闭型保单组,利用历年死亡人数随机向量D,将保单组的未来给付现值随机变量和未来损失现值随机变量表达为某个满秩矩阵和D的乘积,根据D服从多项分布的性质,得到未来损失现值随机向量渐近服从多元正态分布的结果,为分析责任准备金提供了一个新的框架.     

Asymptotic Expansion of the Distribution Density Function for the Sum of Random Variables in the Series Scheme in Large Deviation Zones

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in the Cramer zone and Linnik power zones for the distribution density function of sums of independent random variables in a triangular array scheme. The result was obtained using general Lemma 6.1 of Saulis and Statuleviius in Limit Theorems for Large Deviations (Kluwer, 1991) and joining the methods of characteristic functions and cumulants. The work extends the theory of sums of random variables and in a special case, improves S. A.Book's results on sums of random variables with weights.