STOCHASTIC LIMIT THEORY 随机的极限理论.doc

STOCHASTIC LIMIT THEORY 随机的极限理论Preface 前言Mathematical Symbols and Abbreviations 数学符号和简称Part I : Mathematics 第一部分：数学1. Sets and Numbers 集合与数字1.1 Basic Set Theory 集合理论基础1.2 Corntable Sets 可数集1.3 The Real Continuum 实数连续？1.4 Sequences of Sets集合序列1.5 Classes of Subsets子集类1.6 Sigma Fields -代数2. Limits and Continuity 极限与连续2.1 The Topology of the Real Line 实直线的拓扑2.2 Sequences and Limits 序列与极限2.3 Functions and Continuity 函数与连续2.4 Vector Sequences and Functions 向量序列与函数2.5 Sequences of Functions 函数序列2.6 Summability and Order Relations 可加性与序关系2.7 Arrays 数组、阵列？3. Measure 测度3.1 Measure Spaces 可测空间3.2 The Extension Theorem 推广定理3.3 Non-measurability 不可测性3.4 Product Spaces 积空间3.5 Measurable Thansformations 可测性变换3.6 Borel Functions 博雷尔函数4. Integration 积分4.1 Construction of the Integral 积分的构造4.2 Properties of the Integral 积分的性质4.3 Product Measure and Multiple Integrals 积测量与多维积分？4.4 The Radon-Nikodym Theorem 拉东-尼科迪姆定理5. Metric Spaces 度量空间5.1 Distances and Metrics 距离和度量5.2 Separability and Completeness 可分性和完整性5.3 Examples 举例5.4 Mappings on Metric Spaces 度量空间上的映射5.5 Function Spaces 函数空间6. Topology 拓扑6.1 Topological Spaces 拓扑空间6.2 Countability and Compactness 可数性和紧性6.3 Separation Properties 可分的性质6.4 Weak Topologies 弱拓扑6.5 The Topology of Product Spaces 积空间的拓扑6.6 Embedding and Metrization 嵌入与量化Part II: Probability 第二部分：概率7. Probability Spaces 概率空间7.1 Probability Measures 概率测度7.2 Conditional Probability 条件概率7.3 Independence 独立性7.4 Product Spaces 积空间8. Random Variables 随机变量8.1 Measures on the Line 直线上的测度8.2 Distribution Functions 分布函数8.3 Examples 举例8.4 Multivariate Distributions 多维分布8.5 Independent Random Variables 独立随机变量9. Expectations 期望9.1 Averages and Integrals 平均数与积分9.2 Expectations of Functions of X X的函数的期望9.3 Theorems for the Probabilists Toolbox 概率？的定理9.4 Multivariate Distributions 多维分布9.5 More Theorems for the Toolbox ？的更多定理9.6 Random Variables Depending on a Parameter 依赖单参数的随机变量10. Conditioning 调节？10.1 Conditioning in Product Measures 积度量的调节10.2 Conditioning on a Sigma Field -代数上的制约10.3 Conditional Expectations 条件期望10.4 Some Theorems on Conditional Expectations 条件期望的一些定理10.5 Relationships between Subfields 子域间的相关10.6 Conditional Distributions 条件分布11. Characteristic Functions 特征函数11.1 The Distribution of Sums of Random Variables 随机变量的和的分布11.2 Complex Numbers 复数11.3 The Theory of Characteristic Functions 特征函数的性质11.4 The Inversion Theorem 反演定理11.5 The Conditional Characteristic Function 条件特征函数Part III: Theory of Stochastic Processes 随机过程理论12. Stochastic Processes 随机过程12.1 Basic Ideas and Terminology 基本思想和术语12.2 Convergence of Stochastic Sequences 随机序列的收敛12.3 The Probability Model 概率模型12.4 The Consistency Theorem 一致性定理12.5 Uniform and Limiting Properties 一致性和极限性质12.6 Uniform Integrability 一致可积性13. Dependence 相关13.1 Shift Transformations 移位变换13.2 Independence and Stationarity 独立和平稳性13.3 Invariant Events 不变事件13.4 Ergodicity and Mixing 遍历性和混合13.5 Subfields and Regularity 子域和规律13.6 Strong and Uniform Mixing 强的一致的混合14. Mixing 混合14.1 Mixing Sequences of Random Variables 随机变量的混合序列14.2 Mixing Inequalities 混合不平等性14.3 Mixing in Linear Processes 线性过程中的混合14.4 Sufficient Conditions for Strong and Uniform Mixing 强的一致性混合的充分条件15. Martingales 鞅15.1 Sequential Conditioning 序列的条件15.2 Extensions of the Martingale Concept 鞅概念的推广15.3 Martingale Convergence 鞅收敛15.4 Convergence and the Conditional variances 收敛和条件方差15.5 Martingale Inequalities 鞅不等16. Mixingales 混合性16.1 Definition and Examples 定义和示例16.2 Telescoping Sum Representations 套叠和的表示形式？16.3 Maximal Inequalities 极大不等式16.4 Uniform Square-integrability 一致平方可积性17. Near-Epoch Dependence 近周期相关17.1 Definition and Examples 定义和示例17.2 Near-Epoch Dependence and Mixingales 近周期相关和混合性17.3 Near-Epoch Dependence and Transformations 近周期相关和变换17.4 Approximability 可逼近性Part IV: The Law of Large Numbers 大数定律18. Stochastic Convergence 随机收敛18.1 Almost Sure Convergence 几乎必然收敛18.2 Convergence in Probability 概率的收敛18.3 Transformations and Convergence 变换和收敛18.4 Convergence in Lp Norm Lp范数的收敛18.5 Examples 举例18.6 Laws of Large Numbers 大数定律19. Convergence in Lp-Norm Lp范数的收敛19.1 Weak Laws by Mean-Square Convergence19.2 Almost Sure Convergence by the Method of Subsequences19.3 A Martingale Weak Law19.4 A Mixingale Weak Law19.5 Approximable Processes20. The Strong Law of Large Numbers20