On the convergence of uncertain sequences

Uncertain variables are measurable functions from uncertainty spaces to the set of real numbers. In this paper, a new kind of convergence, convergence uniformly almost surely (convergence uniformly a.s.), is presented. Then, relations between convergence uniformly almost surely and convergence almost surely (convergence a.s.), convergence in measure, convergence in mean, and convergence in distribution are discussed.     

The convergence space of minimal USCO mappings

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all continuous functions is also addressed.      

Convergence of sequences of semi-continuous functions

In this paper we investigate how three well-known modes of convergence for (real-valued) functions are related to one another. In particular, we consider order convergence, pointwise convergence and continuous convergence of sequences of nearly finite normal lower semi-continuous functions. There is a natural comparison to be made between the results we obtain for convergence of sequences of semi-continuous functions, and classic results on the convergence of sequences of measurable functions.     

Complete Moment Convergence for Arrays of Rowwise Widely Orthant Dependent Random Variables

In this paper, complete moment convergence for widely orthant dependent random variables is investigated under some mild conditions. For arrays of rowwise widely orthant dependent random variables, the main results extend recent results on complete convergence to complete moment convergence. These results on complete moment convergence are shown to yield new results on complete integral convergence.     

广义函数Denjoy积分的收敛性问题

本文讨论广义函数De njoy积分的收敛性问题.首先给出了广义Denjoy可积函数空间中强收敛、弱收敛、弱~*收敛和广义函数Denjoy积分收敛的关系;证明拟一致收敛是广义函数Denjoy积分收敛的一个充分必要条件;最后指出了Denjoy可积广义函数列弱~*收敛与强收敛等价当且仅当原函数等度连续.     

On convergence of extremes under power normalization

In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown to coincide with some von Mises conditions.     

Convergence of Polynomial Level Sets

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.

     

Lacunary equi-statistical convergence of positive linear operators

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.      

Convergence on Categories

We introduce and study a concept of a convergence structure on a concrete category. The concept is based on using certain generalized filters for expressing the convergence. Some basic properties of the convergence structures are discussed. In particular, we study convergence separation and convergence compactness and investigate relationships between the convergence structures and the usual closure operators on categories. Dedicated to Professor E. Giuli and Professor G. Preuss on the occasion of their 65th birthdays.     

A note on almost sure uniform and complete convergences of a sequence of random variables

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean.     

Convergence of the Nelder-Mead method

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

     

Normalized convergence in stochastic optimization

A new concept of (normalized) convergence of random variables is introduced. This convergence is preserved under Lipschitz transformations, follows from convergence in mean and itself implies convergence in probability. If a sequence of random variables satisfies a limit theorem then it is a normalized convergent sequence. The introduced concept is applied to the convergence rate study of a statistical approach in stochastic optimization.     

About the statistical uniform convergence

In this work we study the concept of statistical uniform convergence. We generalize some results of uniform convergence in double sequences to the case of statistical convergence. We also prove a basic matrix theorem with statistical convergence.     

On the relations between two types of convergence for convex functions

Theory and applications have shown that there are two important types of convergence for convex functions: pointwise convergence and convergence in a topology induced by the convergence of their epigraphs. We show that these two types of convergence are equivalent on the class of convex functions which are equi-lower semicontinuous. This turns out to be maximal classes of convex functions for which this equivalence can be obtained. We also indicate a number of implications of these results to the convergence of convex sets and the corresponding support functions and to the convergence of the infima of sequences of convex minimization problems.     

Gähler’s neighbourhood condition for convergence approach spaces

We characterize convergence approach spaces that are approach spaces by generalizing a neighbourhood condition from the category of convergence spaces to the category of convergence approach spaces. We also study this condition in the categories of limit tower spaces and probabilistic convergence spaces.     

集值逆上鞅的收敛定理及应用

研究了取值于Ｂａｎａｃｈ空间的集值逆上鞅的收敛性，给出了集值逆上鞅在集列Ｗｉｊｓｍａｎ收敛，弱收敛及Ｋｕｒａｔｏｗｓｋｉ－Ｍｏｓｃｏ收敛意义下的收敛定理，并给出了它们在连续参数集值鞅中的应用。     

Some variations of the Bohman–Korovkin theorem

In this paper we develop the main aspects of the Bohman–Korovkin theorem on approximation of continuous functions with the use of A-statistical convergence and matrix summability method which includes both convergence and almost convergence. Since statistical convergence and almost convergence methods are incompatible we conclude that these methods can be used alternatively to get some approximation results.     

关于随机序列加权和的收敛性的注记

通过对数组加权系数的列截尾,研究了B值随机一致有界序列加权和的收敛性,获得了相应的几乎处处收敛和完全收敛到0的结果.最后建立了B值随机适应序列加权和的矩收敛定理.     

金融功能升级与区域金融发展的动态内生俱乐部收敛

本文在对区域金融收敛特征研究过程中融入金融发展理论的核心思想,并构建包含金融功能升级因素的区域金融发展收敛分析框架,使用金融规模指标、金融结构指标和基于三阶段DEA模型测算的金融效率指标,将动态空间收敛模型和内生门槛收敛模型二者相结合对我国区域金融发展的收敛性和层级特征进行探究,得出结论如下:我国东部地区金融规模表现出收敛的特征,金融结构虽已逐渐匹配于经济结构,但仍呈现一定的组内差异性,金融效率趋于收敛;西部地区金融规模也表现出收敛趋势,但其金融结构与经济结构不相匹配,金融效率呈现出较大的组内差异,都没有表现出收敛的趋势;中部地区则由于需要同时顾及金融规模扩张、金融结构优和金融效率提升而面临着众多困难,其金融发展呈现一定的发散性和低层次特征。金融发展组内异质性导致成员打破传统区域设定,呈现出跨区域向发达省份接近的动态内生俱乐部收敛特征,区域金融发展呈现出不同的层级特征。     

Rates of Ideal Convergence for Approximation Operators

In this paper, we study a general Korovkin-type approximation theory by using the notion of ideal convergence which includes many convergence methods, such as, the usual convergence, statistical convergence, A-statistical convergence, etc. We mainly compute the rate of ideal convergence of sequences of positive linear operators.