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Let $X_1,X_2,\ldots,X_n$ be a sequence of extended negatively dependent random variables with distributions $F_1,F_2,\ldots,F_n$,respectively. Denote by $S_n=X_1+X_2+\cdots+X_n$. This paper establishes the asymptotic relationship for the quantities $\pr(S_n>x)$, $\pr(\max\{X_1,X_2, \ldots,X_n\}>x)$, $\pr(\max\{S_1,S_2$, $\ldots,S_n\}>x)$ and $\tsm_{k=1}^n\pr(X_k>x)$ in the three heavy-tailed cases. Based on this, this paper also investigates the asymptotics for the tail probability of the maximum of randomly weighted sums, and checks its accuracy via Monte Carlo simulations. Finally, as an application to the discrete-time risk model with insurance and financial risks, the asymptotic estimate for the finite-time ruin probability is derived.     

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In this paper, for a kind of risk models with heavy-tailed and delayed claims, we derive the asymptotics of the infinite-time ruin probability and the uniform asymptotics of the finite-time ruin probability. The numerical simulation results are also presented. The results of theoretical analysis and numerical simulation show that the influence of the delay for the claim payment is nearly negligible to the ruin probability when the initial capital and running-time are all large.     

β-Hermite系综关于β-Laguerre系综的逼近

本文研究了β-Hermite系综关于β-Laguerre系综在Kullback-Leibler距离和全变差距离下逼近的问题.利用Pinsker不等式和β-Hermite系综的中心极限定理以及β-Laguerre系综的三阶矩,我们获得了两种距离下逼近的充要条件.此结果推广了文献[4]中的结果.     

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This note deals with an insurance company with multiple lines of business. In the context of heavy-tailed heterogeneous claim amounts with the 1st upper-orthant tail dependence, based on the so-called k-out-of-n ruin set, we can exhibit the Radon measure mu and derive the asymptotic ruin probability for some of all lines business to ruin in a finite time. One numerical example is also presented to illustrate our main results.     

关于β-正则与一致β-正则诱导极限

本文举出反例说明文献［７］中关于一致β－正则的充要条件是错误的，并分别给出了诱导极限为β－正则与一致β－正则的正确的充要条件．     

公式cos(α-β)=cosαcosβ+sinαsinβ的向量解释

在直角坐标系内单位圆上设A (cosα ,sinα) ,B (cosβ ,sinβ)(其中α ,β∈R) ,则OA———→ =(cosα ,sinα) ,OB———→ =(cosβ ,sinβ) .又　 |OA———→| =|OB———→| =1,OA———→·OB———→ =cosαcosβ +sinαsinβ ,cos(α -β) =cos∠BOA =cos〈OA———→ ,OB———→〉 .而OA———→·OB———→ =|OA———→|·|OB———→|cos〈OA———→ ,OB———→〉=cos〈OA———→,OB———→〉=cos(α-β) ,∴　cos(α -β) =cosαcosβ +sinαsinβ .公式cos(α-β)=cosαcosβ+sinαsinβ的向量解释$山…     

β-范空间中l_β(β<1)的渐进等距Copy问题的研究

我们给出了赋β-范空间X包含lβ(β<1)的一个渐进等距copy的定义,并且得到:若一个β-范空间X包含lβ(β<1)的一个渐进等距copy,则在X的闭有界β-凸子集上的非扩张映射没有不动点.还得到了一个β-范空间包含lβ(β<1)的渐进等距copy的某些充分必要条件.注意:本章所有的β满足β<1.     

J_β空间的基本性质以及与l~β的关系

本文给出了赋β范空间J_β(0<β<1)的共轭空间,并介绍了该空间的一些基本性质以及与J~β的关系.     

sin（α±β）＝sinαcosβ±cosαsinβ的几何简证

ｓｉｎ（α±β）＝ｓｉｎαｃｏｓβ±ｃｏｓαｓｉｎβ的几何简证２２５２１１江苏省江都市大桥中学党庆寿本文给出恒等式ｓｉｎ（α±β）＝ｓｉｎαｃｏｓβ±ｃｏｓαｓｎｉβ的一个几何简证．如图１，△ＡＢＣ为Ｒｔ△，∠ＡＣＢ＝９０°，Ｄ在线段ＢＣ的延长线上．...     

独立随机序列最大值的几乎处处极限定理

本文研究了独立随机序列最大值分布的几乎必然收敛性.利用有关协方差的不等式和加权平均,获得独立随机序列最大值的几乎处处极限.将独立同分布随机序列的结论,推广了独立但不同分布的情形.     

负相依随机变量之和的概率大偏差不等式

本文建立了负相依随机变量序列的概率大偏差不等式，并推广了以往文献的结果．     

NA序列部分和的矩完全收敛性

讨论了NA序列部分和的矩完全收敛性,在一定条件下获得了NA序列矩完全收敛的充要条件,显示了矩完全收敛和矩条件之间的关系,将独立同分布随机变量序列矩完全收敛的结果推广到NA序列,得到了与独立随机变量序列情形类似的结果.     

具有不同分布NA随机变量列的强大数律及应用

给出了具有不同分布的NA随机变量列满足的若干强大数律;作为应用,不仅将独立随机变量的一类强极限定理完整的推广到NA随机变量情形,而且关于NA随机变量的一些已有结果可以作为推论得出.     

A strong law of large numbers for fuzzy random variables

A limit of a sequence of fuzzy numbers is defined and its some properties are shown. Based on these concept and properties, an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown.     

The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.     

任意随机变量序列泛函及其随机变换的一类极限定理

采用鞅方法研究对任意随机变量序列普遍成立的强极限定理.并作为推论得到了m阶马氏过程,鞅序列,鞅差序列,独立随机变量序列的一类强极限定理.并把赌博系统的随机变换概念推广到任意随机变量序列的情况,得到任意随机变量序列随机选择与公平比的若干极限定理.     

Conditional independence,conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.     

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.     

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．