概率論和隨機過程(第2版)（簡體書）

作者:(美)凱羅勒夫

凱羅勒夫編著的《概率論和隨機過程》是以作者在Princeton大學和Maryland大學的講義為藍本擴充而成，書中的內容正好可作為《概率論和隨機過程》課程一學年的獨立教材。這對于高年級的本科生、研究生和想要了解本科目基礎知識的科研人員都是相當有用的。本書文筆流暢，其中的概念和相關的結果都是生動豐富，并具有啟發性。每章末都包含難易不等的練習題。這本書已經被作者用作Princeton大學和Maryland高年級本科生和研究生學習該科目的一學期的教程。目次：（第一部分）概率論：隨機變量及其分布；獨立試驗序列；勒貝格積分和數學期望；條件概率和期望；具有有限數狀態的馬爾科夫鏈；大數定理；測度的弱收斂；特征函數；極限定理；幾個有趣的問題;（第二部分）隨機過程：基本概念；條件期望和鞅；有限狀態空間的馬爾科夫鏈；廣泛意義上的平穩隨機過程；嚴格平穩隨機過程；廣義隨機過程；布朗運動；馬爾科夫過程和馬爾科夫族；隨機積分和Ito公式；隨機微分方程；Gibbs隨機域。

Part Ⅰ Probability Theory
1 Random Variables and Their Distributions
1.1 Spaces of Elementary Outcomes， a-Algebras， and Measures
1.2 Expectation and Variance of Random Variables on a Discrete Probability Space
1.3 Probability of a Union of Events
1.4 Equivalent Formulations of a-Additivity， Borel a-Algebras and Measurability
1.5 Distribution Functions and Densities
1.6 Problems
2 Sequences of Independent Trials
2.1 Law of Large Numbers and Applications
2.2 de Moivre-Laplace Limit Theorem and Applications
2.3 Poisson Limit Theorem.
2.4 Problems
3 Lebesgue Integral and Mathematical Expectation
3.1 Definition of the Lebesgue Integral
3.2 Induced Measures and Distribution Functions
3.3 Types of Measures and Distribution Functions
3.4 Remarks on the Construction of the Lebesgue Measure
3.5 Convergence of Functions， Their Integrals， and the Fubini Theorem
3.6 Signed Measures and the R，adon-Nikodym Theorem
3.7 Lp Spaces
3.8 Monte Carlo Method
3.9 Problems
4 Conditional Probabilities and Independence
4.1 Conditional Probabilities
4.2 Independence of Events， Algebras， and Random Variables
4.3
4.4 Problems
5 Markov Chains with a Finite Number of States
5.1 Stochastic Matrices
5.2 Markov Chains
5.3 Ergodic and Non-Ergodic Markov Chains
5.4 Law of Large Numbers and the Entropy of a Markov Chain
5.5 Products of Positive Matrices
5.6 General Markov Chains and the Doeblin Condition
5.7 Problems
6 Random Walks on the Lattice Zd
6.1 Recurrent and Transient R，andom Walks
6.2 Random Walk on Z and the Refiection Principle
6.3 Arcsine Law
6.4 Gambler's Ruin Problem
6.5 Problems
7 Laws of Larze Numbers
7.1 Definitions， the Borel-Cantelli Lemmas， and the Kolmogorov Inequality
7.2 Kolmogorov Theorems on the Strong Law of Large Numbers
7.3 Problems
8 Weak Converaence of Measures
8.1 Defnition of Weak Convergence
8.2 Weak Convergence and Distribution Functions
8.3 Weak Compactness， Tightness， and the Prokhorov Theorem
8.4 Problems
9 Characteristic Functions
9.1 Definition and Basic Properties
9.2 Characteristic Functions and Weak Convergence
9.3 Gaussian Random Vectors
9.4 Problems
10 Limit Theorems
10.1 Central Limit Theorem， the Lindeberg Condition
10.2 Local Limit Theorem
10.3 Central Limit Theorem and Renormalization GrOUD Theorv
10.4 Probabilities of Large Deviations
……
Part Ⅱ Random Processes
Index