Probability Theory
Probability Theory
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DOI：
10.1090/cln/007
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
S. Varadhan
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文献类型：
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作者：
S. Varadhan
本卷介绍了在第一年的研究生课程在柯朗数学科学研究所所涵盖的概率论主题。必要的背景材料在测量理论的发展，包括标准的主题，如扩展定理，建设措施，整合，产品空间，Radon-Nikodym定理，和条件期望。在书的第一部分，特征函数被介绍，其次是弱收敛的概率分布的研究。然后证明了独立随机变量和的弱、强极限定理，包括弱、强大数定律、中心极限定理、重对数定律和Kolmogorov三级数定理。第一部分给出了一致无穷小独立随机变量和的无穷可分分布和极限定理。第二部分的书主要涉及依赖随机变量，特别是鞅和马尔可夫链。主题包括离散参数鞅和Doob不等式的标准结果。马尔可夫链中的标准主题的治疗，即，瞬态，零和积极的复发。给出了一系列不同的例子来证明鞅和马尔可夫链之间的联系。书中涉及的其他主题包括平稳高斯过程，遍历定理，动态规划，最佳停止和过滤。包括大量的例子和练习。这本书是一个合适的文本一年级研究生课程的概率。
This volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences. The necessary background material in measure theory is developed, including the standard topics, such as extension theorem, construction of measures, integration, product spaces, Radon-Nikodym theorem, and conditional expectation. In the first part of the book, characteristic functions are introduced, followed by the study of weak convergence of probability distributions. Then both the weak and strong limit theorems for sums of independent random variables are proved, including the weak and strong laws of large numbers, central limit theorems, laws of the iterated logarithm, and the Kolmogorov three series theorem. The first part concludes with infinitely divisible distributions and limit theorems for sums of uniformly infinitesimal independent random variables. The second part of the book mainly deals with dependent random variables, particularly martingales and Markov chains. Topics include standard results regarding discrete parameter martingales and Doob's inequalities. The standard topics in Markov chains are treated, ie, transience, and null and positive recurrence. A varied collection of examples is given to demonstrate the connection between martingales and Markov chains. Additional topics covered in the book include stationary Gaussian processes, ergodic theorems, dynamic programming, optimal stopping, and filtering. A large number of examples and exercises is included. The book is a suitable text for a first-year graduate course in probability.