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This paper considers the asymptotics of randomly weighted sums and their maxima, where the increments {X_i,i\geq1\} is a sequence of independent, identically distributed and real-valued random variables and the weights {\theta_i,i\geq1\} form another sequence of non-negative and independent random variables, and the two sequences of random variables follow some dependence structures. When the common distribution F of the increments belongs to dominant variation class, we obtain some weakly asymptotic estimations for the tail probability of randomly weighted sums and their maxima. In particular, when the F belongs to consistent variation class, some asymptotic formulas is presented. Finally, these results are applied to the asymptotic estimation for the ruin probability.     

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Let p_M(t,x,y) be the minimal heat kernel of a d-dimenional compact Riemannian manifold M for any time t\in(0,1] and x,y\in M. Using the horizontal Brown bridge on M, we prove that, for any nonnegative integers n and m, there is a constant C depending on n,m and the manifold M, such that |\nabla^n_x\nabla^m_y\ln p_M(t,x,y)|\leq C[d(x,y)/t+1/\sqrt{t}\,]^{n+m}$, which generalizes the conclusion of the higher derivatives of the logarithmic heat kernel \ln p_M(t,x,y) about single variable in \ncite{1}.     

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In this paper, we introduce the definitions of geometric strongly ergodic, strongly ergodic and weakly ergodic for continuous-state Markov chains, then we give a primary proof of equivalence of the ergodicities for continuous-state Markov chains.     

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This paper studies the estimation of change point in mean and variance function of a non-parametric regression model based on kernel estimation and wavelet method. First, kernel estimation of mean function is developed and it is used to estimate the position and jump size of mean change. Second, wavelet methods are applied to derive the variance estimator which is used to estimate the location and jump size of the change point in variance. The asymptotic properties of these estimators are proved. Finally, the results from a numerical simulations and comparison study show that validate the effectiveness of our method.     

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VaR风险度量在金融、保险中有重要的应用. 本文建立了贝叶斯模型, 在某种损失函数下研究了VaR风险度量的贝叶斯估计. 证明了指数-伽马分布下贝叶斯估计的强相合性和渐近正态性, 最后利用数值模拟的方法验证了不同样本容量下估计的收敛速度.     

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??How to solve the inference problem of candidate database web surveys is an urgent problem to be solved in the development of web survey. In order to solve this problem, the inference method of non-probability sampling based on superpopulation pseudo design and the combined sample is proposed. A superpopulation model is firstly built up to construct pseudo weights for a survey sample of the web candidate database. The estimator of the population mean is then computed according to the combined sample composed of the survey sample of the web candidate database and a probability sample. The variance estimator of the population mean estimator is lastly derived according to the variance estimation theory of the superpopulation model. The Bootstrap and Jackknife methods are also used to compute the variance estimator. And all these variance estimation methods are compared. The research results show that the population mean estimator based on superpopulation pseudo design and the combined sample is better, and has higher efficiency than the estimator only using the probability sample and the weighted estimator only using the survey sample of the web candidate database. The variance estimator computed by using the VM1, VM2 and VM3 method are relatively better.     

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Let, be two independent,
-dimensional sub-fractional Brownian motions with respective indices.
Assume. Our principal results are the necessary and sufficient condition for the
existence and smoothness of the collision local time and the intersection local time of
and through chaos expansion and elementary inequalities.     

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There're about 10^{11} neurons in the human brain.Through the synaptic junction, neurons have formed a highly complex network.And it is really important to figure out the information expressed in the network, which will contribute to the resolution of the prevention and diagnosis of cognitive disorder of human beings. This paper uses the schizophrenia and healthy controlled subjects' fMRI data to construct the brain network model, in order to explores abnormal topological properties of schizophrenics' brain network based on graph theory. When studying the human brain network information traditionally by the basement of graph theory, it's all assure that the human brain network model has invariance, so it takes the whole period of time series data in constructing human network model, which is a kind static network. However, it's hard to ensure this because of the nonstationarity of fMRI functional time series data. Thus, when constructing human brain network model, we should take its time-variation into consideration, then construct a dynamic brain network. We can explore the brain network information better. In this research, we segment the time series data, using time windows, to constructing dynamic brain network model, then analyze it combined with the knowledge of graph theory, thereby reducing effects that the nonstationarity of fMRI functional time series data will have. Comparing dynamic brain network of the schizophrenic patients with normal controls subjects' in different level, the results show that there are difference in single node property, group network property of schizophrenic patients and normal control subjects' whole brain dynamic functional connectivity network. The discovery of these difference in network topological properties has provide new clues for the further study on the pathological mechanism of schizophrenia.     

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In order to solve the problem of testing multidimensional diffusion models, we develop a test statistic based on Multi-Dimensional Tail Condition Expectations (CTEs). Although it is almost impossible to estimate the transition density matrix of a multidimensional diffusion model directly, the transition density of each component can be estimated and each component can be combined by the CTE to establish a true multidimensional statistics. Finally, the performance of the test is evaluated through simulation.     

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本文证明了copula , 生存copula , 对偶copula 和伴随copula 关于copula的复合运算构成一个四元群, 给出了当某个单点值给定时它们的最优上下界. 计算了, 时copula最优上下界的宽窄度, 并与时的宽窄度进行了比较.     

独立的无界随机变量和的概率不等式

研究了在概率空间(Ω,T,P)上,独立的无界随机变量和尾部概率不等式,提出了一种用切割原始概率空间(Ω,T,P)的新型方法去处理独立的无界随机变量和。给出了独立的无界随机变量和的指数型概率不等式。作为结果的应用,一些有趣的例子被给出。这些例子表明:文中提出的方法和结果对研究独立的无界随机变量和的大样本性质是十分有用的。     

Banach空间的型与独立Banach空间值随机变量序列的不等式

本文建立了独立Ｂ值随机变量序列部分和的极大值函数Ｓ＊与ｐ方根函数Ｓ（ｐ，Ｘ）的分布函数不等式与矩不等式，讨论了这些不等式的成立与Ｂａｎａｃｈ空间的ｐ型与ｑ余型的关系，同时给出了与Ｈｉｌｂｅｒｔ空间同构的Ｂａｎａｃｈ空间的特征。     

A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Let {X _i, 1in} be a negatively associated sequence, and let {X* _i , 1in} be a sequence of independent random variables such that X* _i and X _i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef( ⁿ _i=1 X _i)Ef( ⁿ _i=1 X* _i ) for any convex function f on R ¹ and that Ef(max_1kn ⁿ _i=k X _i)Ef(max_1kn ^k _i=1 X* _i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.     

负相协随机变量的指数不等式

该文给出了一些负相协随机变量的指数不等式.这些不等式改进了由Jabbari和Azarnoosh^[4]及Oliveira^[7] 所得到的相应的结果.利用这些不等式对协方差系数为几何下降情形, 获得了强大数律的收敛速度为n^-1/2(log log n)^1/2(log n)^2.这个收敛速度接近独立随机变量的重对数律的收敛速度, 而Jabbari和Azarnoosh^[4]在上述情形下得到的收敛速度仅仅为n^-1/3(log n)^5/3.     

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In this note we discuss uniform integrability of random variables. In a probability space, we introduce two new notions on uniform integrability of random variables, and prove that they are equivalent to the classic one. In a sublinear expectation space, we give de La Vall\'{e}e Poussin criterion for the uniform integrability of random variables and do some other discussions.     

关于概率空间与次线性期望空间中的随机变量一致可积性的一个注记（英文）

在这篇注记中我们讨论随机变量的一致可积性.在概率空间中,我们引进了随机变量一致可积性的两个新的定义,并证明了他们与经典定义等价.在次线性期望空间中,我们给出了随机变量一致可积性的德拉瓦利普桑准则,并作了一些其它讨论.     

两两独立序列加权和的L~r收敛性

讨论了两两独立随机变量列加权和在满足r(1≤r<2)阶Ces`aro一致可积条件下的Lr收敛性,获得了与独立情形一致的结果.用相似的方法,对于其它相依或混合序列(如两两NQD列,φ-混合序列,ρ-混合序列)也有相同的结果.     

两个变量的Hardy型积分不等式

建立了两个独立变量函数的新的Hardy型积分不等式.     

Some Exponential Inequalities for Negatively Orthant Dependent Random Variables

Acta Mathematicae Applicatae Sinica, English Series - In the paper, we establish some exponential inequalities for non-identically distributed negatively orthant dependent (NOD, for short) random...     

对称随机变量部分和的概率不等式

在对称随机变量分布函数关于原点的值大于或等于二分之一的基础上,阐明对称随机变量的部分和仍是对称随机变量,进一步,给出关于对称随机变量序列部分和的概率不等式.