概率論和隨機(jī)過程

內(nèi)容概要

　　This book is primarily based on a one-year course that has
been taught for a number of years at Princeton University to
advanced undergraduate and graduate students. During the last year
a similar course has also been taught at the University of
Maryland.
　　We would like to express our thanks to Ms. Sophie Lucas and Prof.
Rafael Herrera who read the manuscript and suggested many
corrections. We are particularly grateful to Prof. Boris Gurevich
for making many important sug-gestions on both the mathematical
content and style.
　　While writing this book， L. Koralov was supported by a National
Sci-ence Foundation grant （DMS-0405152）. Y. Sinai was supported by
a National Science Foundation grant （DMS-0600996）.

作者簡(jiǎn)介

作者:(美)凱羅勒夫

書籍目錄

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

章節(jié)摘錄

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《概率論和隨機(jī)過程(第2版)》是以作者在Princeton大學(xué)和Maryland大學(xué)的講義為藍(lán)本擴(kuò)充而成，書中的內(nèi)容正好可作為《概率論和隨機(jī)過程》課程一學(xué)年的獨(dú)立教材。這對(duì)于高年級(jí)的本科生、研究生和想要了解本科目基礎(chǔ)知識(shí)的科研人員都是相當(dāng)有用的。全書文筆流暢，其中的概念和相關(guān)的結(jié)果都是生動(dòng)豐富，并具有啟發(fā)性。每章末都包含難易不等的練習(xí)題。此書已經(jīng)被作者用作Princeton大學(xué)和Maryland高年級(jí)本科生和研究生學(xué)習(xí)該科目的一學(xué)期的教程。目次：（第一部分）概率論：隨機(jī)變量及其分布；獨(dú)立試驗(yàn)序列；勒貝格積分和數(shù)學(xué)期望；條件概率和期望；具有有限數(shù)狀態(tài)的馬爾科夫鏈；大數(shù)定理；測(cè)度的弱收斂；特征函數(shù)；極限定理；幾個(gè)有趣的問題；（第二部分）隨機(jī)過程：基本概念；條件期望和鞅；有限狀態(tài)空間的馬爾科夫鏈；廣泛意義上的平穩(wěn)隨機(jī)過程；嚴(yán)格平穩(wěn)隨機(jī)過程；廣義隨機(jī)過程；布朗運(yùn)動(dòng)；馬爾科夫過程和馬爾科夫族；隨機(jī)積分和Ito公式；隨機(jī)微分方程；Gibbs隨機(jī)域。

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