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Interpolation of Lorentz martingale spaces

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Science China Mathematics 2012年第9期55卷 1951-1959页

作者： REN YanBo guo tiexin School of Mathematics and Statistics Henan University of Science and Technology Luoyang 471003 China LMIB and School of Mathematics and Systems Science Beihang University Beijing 100191 China

In this paper,we apply function parameters,introduced by Persson,to real interpolation of Lorentz martingale *** new interpolation theorems concerning Lorentz martingale spaces are *** results that we obtain generaliz... 详细信息

关键词： martingale Lorentz space interpolation function parameter

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Random duality

Science China Mathematics 2009年第10期52卷 2084-2098页

作者： guo tiexin CHEN XinXiang Key Laboratory of Information Mathematics and Behavior Ministry of EducationDepartment of MathematicsBeihang UniversityBeijing 100191China School of Mathematical Sciences Xiamen UniversityXiamen 361005China

The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(μ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.

关键词： random duality weakly continuous canonical module homomorphism random compatible structure random bipolar theorem 46A20 46A16 46H05 46H25

The relations among the three kinds of conditional risk measures

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Science China Mathematics 2014年第8期57卷 1753-1764页

作者： guo tiexin ZHAO ShiEn ZENG XiaoLin School of Mathematics and Statistics Central South University Elementary Educational College Capital Normal University College of Mathematics and Statistics Chongqing Technology and Business University

Let（Ω , E, P） be a probability space, F a sub-σ-algebra of E, L^p（E）（1 p ＋∞） the classical function space and LF^p（E） the L^0（F）-module generated by L^p（E）, which can be made into a random normed module in a natural way. Up to the present time, there are three kinds of conditional risk measures, whose model spaces are L^∞（E）, L^p（E）（1 p ＋∞） and LF^p（E）（1 p ＋∞） respectively, and a conditional convex dual representation theorem has been established for each kind. The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems. We first establish the relation between L^p（E） and LF^p（E）, namely LF^p（E） = Hcc（L^p（E））, which shows that LF^p（E）is exactly the countable concatenation hull of L^p（E）. Based on the precise relation, we then prove that every L^0（F）-convex L^p（E）-conditional risk measure（1 p ＋∞） can be uniquely extended to an L^0（F）-convex LF^p（E）-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of L^p-conditional risk measures can be incorporated into that of LF^p（E）-conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L^0-convex conditional risk measures.

关键词： random normed module countable concatenation property L^∞(E)-conditional risk measure L^p(E)-conditional risk measure(1≤ p 〈 +∞) LF^p(E)-conditional risk measure(1 ≤p≤ +∞) extension

L^0-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN L^0(F,B)

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Acta Mathematica Scientia 2020年第2期40卷 457-469页

作者： tiexin guo Erxin ZHANG Yachao WANG George YUAN School of Mathematics and Statistics Central South UniversityChangsha 410083China Centre for Financial Engineering Soochow UniversitySuzhou 215006China

Let(B,||·||)be a Banach space,(?,F,P)a probability space,and L^0(F,B)the set of equivalence classes of strong random elements(or strongly measurable functions)from(?,F,P)to(B,||·||).It is well known that L^0(F,B)becomes a complete random normed module,which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator *** V be a closed convex subset of B and L^0(F,V)the set of equivalence classes of strong random elements from(?,F,P)to *** central purpose of this article is to prove the following two results:(1)L^0(F,V)is L^0-convexly compact if and only if V is weakly compact;(2)L^0(F,V)has random normal structure if V is weakly compact and has normal *** an application,a general random fixed point theorem for a strong random nonexpansive operator is given,which generalizes and improves several well known *** hope that our new method,namely skillfully combining measurable selection theorems,the theory of random normed modules,and Banach space techniques,can be applied in the other related aspects.

关键词： Complete random normed modules fixed point theorem L^0-convex compactness random normal structure random nonexpansive operators

METRICS,PROBABILISTIC METRICS AND RANDOM METRICS

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Systems Science and Mathematical Sciences 1995年第2期8卷 182-186页

作者： guo tiexin(Department of Mathematics,Xiamen University, Fujian Xiamen 361005,China) 厦门大学数学系福建厦门

ＭＥＴＲＩＣＳ，ＰＲＯＢＡＢＩＬＩＳＴＩＣＭＥＴＲＩＣＳＡＮＤＲＡＮＤＯＭＭＥＴＲＩＣＳＧＵＯＴｉｅｘｉｎ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉａｍｅｎＵｎｉｖｅｒｓｉｔｙ，ＦｕｊｉａｎＸｉａｍｅｎ３... 详细信息

ＭＥＴＲＩＣＳ，ＰＲＯＢＡＢＩＬＩＳＴＩＣＭＥＴＲＩＣＳＡＮＤＲＡＮＤＯＭＭＥＴＲＩＣＳＧＵＯＴｉｅｘｉｎ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉａｍｅｎＵｎｉｖｅｒｓｉｔｙ，ＦｕｊｉａｎＸｉａｍｅｎ３６１００５，Ｃｈｉｎａ）Ａｂｓｔｒ...

关键词： Largest weak t-norm isometrical isomorphism metric space-system probabilistic metric space-system random metric space-system

随机自反随机赋范模上的平均遍历定理(英文)

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数学进展 2012年第1期41卷 21-30页

作者：张霞郭铁信北京航空航天大学数学与系统科学学院北京100191

本文对随机自反随机赋范模上的几乎处处幂有界的随机线性算子证明了一个平均遍历定理,这一定理推广和改进了几个已知的重要的结果.

关键词：随机赋范模几乎处处幂有界的随机线性算子随机自反平均遍历定理

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-mulian theorem in complete random normed modules to stratification structure

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Science China Mathematics 2008年第9期51卷 1651-1663页

作者： guo tiexin Key Laboratory of Information Mathematics and Behavior of Ministry of Education, Department of Mathemat-ics, Beihang University, Beijing 100083, China 1. Key Laboratory of Information Mathematics and Behavior of Ministry of Education Department of Mathematics Beihang University Beijing 100083 China

Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (Ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A : there exists an element p in S such that Xp(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S*(1) = {f ∈ S* : Xf* 1} of the random conjugate space (S*,X*) of (S,X) is compact under the random weak star topology on (S*,X*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {An : n ∈ N} of at most countably many μ-atoms from E∩A such that E =∪n∞=1 An and for each element F in E∩A, there is an H in the σ-algebra generated by {An : n ∈ N} satisfying μ(F △H) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-mulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S : Xp 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E∩A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed module

关键词： random normed module random reflexivity random weak star compactness random weak compactness stratification structure

随机局部凸模中的Krein-Milman定理及其应用

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中国科学：数学 2023年第12期53卷 1667-1684页

作者：郭铁信王亚超唐艳中南大学数学与统计学院长沙410083 四川大学数学学院成都610064 重庆工商大学数学与统计学院重庆400067

本文首先深入研究随机局部凸模中的稳定紧集,证明它关于(ε,λ)-拓扑Tε,λ是完备的,并给出它的一个简明的特征,即一个σ-稳定集是稳定紧的当且仅当它的每个具有有限交性质的由σ-稳定的Tε,λ-闭子集组成的σ-稳定族必有非空交.在此基... 详细信息

本文首先深入研究随机局部凸模中的稳定紧集,证明它关于(ε,λ)-拓扑Tε,λ是完备的,并给出它的一个简明的特征,即一个σ-稳定集是稳定紧的当且仅当它的每个具有有限交性质的由σ-稳定的Tε,λ-闭子集组成的σ-稳定族必有非空交.在此基础上,对定义在稳定紧集上的σ-稳定的、真的、下半连续的L^(0)-值函数给出相应的Weierstrass定理,并由此证明一个稳定紧的L^(0)-凸集必为L^(0)-凸紧的.然后,对L^(0)-凸集引进L^(0)-端点的概念并对L^(0)-凸紧集证明相应的Krein-Milman定理,同时给出这个推广的Krein-Milman定理与经典的Krein-Milman定理的某些有趣的比较与联系.最后,作为应用,证明定义在L^(0)-凸紧集上的真下半连续L^(0)-拟凸函数f必达到最小值.进一步地,如果f还是L^(0)-仿射的,那么f的最小值也可以在L^(0)-端点达到.

关键词：随机局部凸模稳定紧集 L^(0)-凸紧集 L^(0)-端点 Krein-Milman定理

随机凸分析(Ⅱ):L^0-准桶的随机局部凸模中的连续性和次可微性定理

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中国科学：数学 2015年第5期45卷 647-662页

作者：郭铁信赵世恩曾小林中南大学数学与统计学院长沙410083 首都师范大学初等教育学院北京100048 重庆工商大学数学与统计学院重庆400067

本文继续研究随机凸分析.首先,引入L0-准桶模的概念;接着,在赋予局部L0-凸拓扑的随机局部凸模的框架下,为了建立随机局部凸模为L0-准桶模的特征,本文发展了随机对偶理论,尤其是证明用于条件风险度量的模途径中的模型空间LpF(E)是L0-准桶... 详细信息

本文继续研究随机凸分析.首先,引入L0-准桶模的概念;接着,在赋予局部L0-凸拓扑的随机局部凸模的框架下,为了建立随机局部凸模为L0-准桶模的特征,本文发展了随机对偶理论,尤其是证明用于条件风险度量的模途径中的模型空间LpF(E)是L0-准桶的,这形成本文最困难的部分.最后,本文证明L0-准桶的随机局部凸模上的真下半连续L0-凸函数的连续性和次可微性定理.因此,本文的主要结果可以很好地适用于L0-凸条件风险度量的连续性和次可微性的研究.

关键词：随机局部凸模局部L0-凸拓扑随机共轭空间真L0-凸下半连续函数连续性次可微性