Probability Theory A Comprehensive Course 2nd edition by Achim Klenke – Ebook PDF Instant Download/Delivery: 1447153610 , 9781447153610
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ISBN 10: 1447153610
ISBN 13: 9781447153610
Author: Achim Klenke
This second edition of the popular textbook contains a comprehensive course in modern probability theory, covering a wide variety of topics which are not usually found in introductory textbooks, including: • limit theorems for sums of random variables • martingales • percolation • Markov chains and electrical networks • construction of stochastic processes • Poisson point process and infinite divisibility • large deviation principles and statistical physics • Brownian motion • stochastic integral and stochastic differential equations. The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in probability theory. This second edition has been carefully extended and includes many new features. It contains updated figures (over 50), computer simulations and some difficult proofs have been made more accessible. A wealth of examples and more than 270 exercises as well as biographic details of key mathematicians support and enliven the presentation. It will be of use to students and researchers in mathematics and statistics in physics, computer science, economics and biology.
Probability Theory A Comprehensive Course 2nd Table of contents:
Chapter 1: Basic Measure Theory
1.1 Classes of Sets
1.2 Set Functions
1.3 The Measure Extension Theorem
1.4 Measurable Maps
1.5 Random Variables
Chapter 2: Independence
2.1 Independence of Events
2.2 Independent Random Variables
2.3 Kolmogorov’s 0-1 Law
2.4 Example: Percolation
Uniqueness of the Infinite Open Cluster*
Chapter 3: Generating Functions
3.1 Definition and Examples
3.2 Poisson Approximation
3.3 Branching Processes
Chapter 4: The Integral
4.1 Construction and Simple Properties
4.2 Monotone Convergence and Fatou’s Lemma
4.3 Lebesgue Integral Versus Riemann Integral
Chapter 5: Moments and Laws of Large Numbers
5.1 Moments
5.2 Weak Law of Large Numbers
5.3 Strong Law of Large Numbers
Entropy and Source Coding Theorem*
5.4 Speed of Convergence in the Strong LLN
5.5 The Poisson Process
Chapter 6: Convergence Theorems
6.1 Almost Sure and Measure Convergence
6.2 Uniform Integrability
6.3 Exchanging Integral and Differentiation
Chapter 7: Lp-Spaces and the Radon-Nikodym Theorem
7.1 Definitions
7.2 Inequalities and the Fischer-Riesz Theorem
7.3 Hilbert Spaces
7.4 Lebesgue’s Decomposition Theorem
7.5 Supplement: Signed Measures
7.6 Supplement: Dual Spaces
Chapter 8: Conditional Expectations
8.1 Elementary Conditional Probabilities
8.2 Conditional Expectations
8.3 Regular Conditional Distribution
Chapter 9: Martingales
9.1 Processes, Filtrations, Stopping Times
9.2 Martingales
9.3 Discrete Stochastic Integral
9.4 Discrete Martingale Representation Theorem and the CRR Model
Chapter 10: Optional Sampling Theorems
10.1 Doob Decomposition and Square Variation
10.2 Optional Sampling and Optional Stopping
10.3 Uniform Integrability and Optional Sampling
Chapter 11: Martingale Convergence Theorems and Their Applications
11.1 Doob’s Inequality
11.2 Martingale Convergence Theorems
11.3 Example: Branching Process
Chapter 12: Backwards Martingales and Exchangeability
12.1 Exchangeable Families of Random Variables
Heuristic for the Structure of Exchangeable Families
12.2 Backwards Martingales
12.3 De Finetti’s Theorem
Chapter 13: Convergence of Measures
13.1 A Topology Primer
13.2 Weak and Vague Convergence
13.3 Prohorov’s Theorem
13.4 Application: A Fresh Look at de Finetti’s Theorem
Chapter 14: Probability Measures on Product Spaces
14.1 Product Spaces
14.2 Finite Products and Transition Kernels
14.3 Kolmogorov’s Extension Theorem
14.4 Markov Semigroups
Chapter 15: Characteristic Functions and the Central Limit Theorem
15.1 Separating Classes of Functions
15.2 Characteristic Functions: Examples
15.3 Lévy’s Continuity Theorem
15.4 Characteristic Functions and Moments
15.5 The Central Limit Theorem
15.6 Multidimensional Central Limit Theorem
Chapter 16: Infinitely Divisible Distributions
16.1 Lévy-Khinchin Formula
16.2 Stable Distributions
Symmetric Stable Distributions
General Stable Distributions
Convergence to Stable Distributions
Chapter 17: Markov Chains
17.1 Definitions and Construction
17.2 Discrete Markov Chains: Examples
17.3 Discrete Markov Processes in Continuous Time
17.4 Discrete Markov Chains: Recurrence and Transience
17.5 Application: Recurrence and Transience of Random Walks
17.6 Invariant Distributions
17.7 Stochastic Ordering and Coupling
Chapter 18: Convergence of Markov Chains
18.1 Periodicity of Markov Chains
18.2 Coupling and Convergence Theorem
18.3 Markov Chain Monte Carlo Method
Metropolis Algorithm
Gibbs Sampler
Perfect Sampling
18.4 Speed of Convergence
Chapter 19: Markov Chains and Electrical Networks
19.1 Harmonic Functions
19.2 Reversible Markov Chains
19.3 Finite Electrical Networks
19.4 Recurrence and Transience
19.5 Network Reduction
Reduced Network
Step-by-Step Reduction of the Network
Application to Example 19.32
Alternative Solution
19.6 Random Walk in a Random Environment
Chapter 20: Ergodic Theory
20.1 Definitions
20.2 Ergodic Theorems
20.3 Examples
20.4 Application: Recurrence of Random Walks
20.5 Mixing
20.6 Entropy
Chapter 21: Brownian Motion
21.1 Continuous Versions
21.2 Construction and Path Properties
21.3 Strong Markov Property
21.4 Supplement: Feller Processes
21.5 Construction via L2-Approximation
Lévy Construction of Brownian Motion
Brownian Motion and White Noise
21.6 The Space C([0,infty))
21.7 Convergence of Probability Measures on C([0,infty))
21.8 Donsker’s Theorem
21.9 Pathwise Convergence of Branching Processes*
21.10 Square Variation and Local Martingales
Chapter 22: Law of the Iterated Logarithm
22.1 Iterated Logarithm for the Brownian Motion
22.2 Skorohod’s Embedding Theorem
Supplement: Proof of Remark 22.6
22.3 Hartman-Wintner Theorem
Chapter 23: Large Deviations
23.1 Cramér’s Theorem
23.2 Large Deviations Principle
23.3 Sanov’s Theorem
23.4 Varadhan’s Lemma and Free Energy
Chapter 24: The Poisson Point Process
24.1 Random Measures
24.2 Properties of the Poisson Point Process
24.3 The Poisson-Dirichlet Distribution*
The Chinese Restaurant Process
Chapter 25: The Itô Integral
25.1 Itô Integral with Respect to Brownian Motion
25.2 Itô Integral with Respect to Diffusions
25.3 The Itô Formula
25.4 Dirichlet Problem and Brownian Motion
25.5 Recurrence and Transience of Brownian Motion
Chapter 26: Stochastic Differential Equations
26.1 Strong Solutions
26.2 Weak Solutions and the Martingale Problem
26.3 Weak Uniqueness via Duality
Notation Index
References
Name Index
Subject Index
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Tags: Achim Klenke, Probability Theory, Comprehensive Course