论文标题:统计收敛的测度理论
Measure Theory of Statistical Convergence
论文作者
论文导师 程立新,论文学位 硕士,论文专业 基础数学
论文单位 厦门大学,点击次数 75,论文页数 42页File Size1781K
2007-05-01论文网 http://www.lw23.com/lunwen_52140992/
statistical convergence;; statistical measure;; sub differential
早在1951年，H．Fast[43]就引入了统计收敛的定义，之后，出现了一系列的相关文章(1-20,45-52,48,49,74,74,21-42,53-56,58-64,66,67,72,73,76-98,100-102,104-131)对统计收敛做了进一步的探索与研究．随着统计收敛理论的发展，建立统计收敛测度理论的问题也逐渐成为一个核心问题，因为一种合理的理论不仅是把各种统计收敛统一起来的原则，而且是统计收敛通向测度理论，积分理论，概率论和数理统计的桥梁．基于这个原因，证明了以下结论： 1定义在由M的所有子集生成的σ-代数(?)上的所有有限可加概率测度的表示理论． 2每个如1中的有限可加概率测度都可以唯一的分解为一个可列可加概率测度和一个统计测度(即一个有限可加概率测度μ，对任意的单点集{k}有μ(k)=0)的凸组合． 本文同样证明了许多经典统计测度的性质，例如： 3由所有经典统计测度组成的集合(?)在(?)上赋予逐点收敛的拓扑就成为紧凸的Hausdorff空间． 4每—个经典统计测度都是连续型的(所以是缺原子的)． 5对N中的任意的集合，每—类特殊的统计测度都满足complementationminmax原则． 6每—类统计收敛都可以统一为统计测度的收敛．
The notion of statstical convergence was introduced by Fast [43] in 1951. From then on, statistical convergence had been investigated and developed in a sequence of articals(see, for instance, [1-20,45-52,48,49,74,75,21-42,53-56,58-64,66,67,72,73,76-98,100-102,104-131]. with the development of statistical convergence, the question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the study of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper shows many theorems as follows. 1 For all finitely additive probability measures defined on theσ-algebra (?) of all subsets of N. 2 Proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measureμwithμ(k) = 0 for all singletons {k} ). This paper also shows that classical statistical measures have many nice properties, such as: 3 The set (?) of all such measures endowed with the topology of point-wise convergence on (?) forms a compact convex Hausdorff space. 4 Every classical statistical measure is of continuity type (hence, atom-less). 5 Every specific class of statistical measures fits a complementation minimax rule for every subset in N. 6 This paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.
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