Ebook: Probability Theory: A Comprehensive Course

This second edition of the popular textbook contains a comprehensive course in modern probability theory. Overall, probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us in understanding magnetism, amorphous media, genetic diversity and the perils of random developments at financial markets, and they guide us in constructing more efficient algorithms.

To address these concepts, the title covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:

• 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.

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This second edition of the popular textbook contains a comprehensive course in modern probability theory. Overall, probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us in understanding magnetism, amorphous media, genetic diversity and the perils of random developments at financial markets, and they guide us in constructing more efficient algorithms.

To address these concepts, the title covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:

• 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.

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This second edition of the popular textbook contains a comprehensive course in modern probability theory. Overall, probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us in understanding magnetism, amorphous media, genetic diversity and the perils of random developments at financial markets, and they guide us in constructing more efficient algorithms.

To address these concepts, the title covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:

• 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.

Content:
Front Matter....Pages I-XII
Basic Measure Theory....Pages 1-45
Independence....Pages 47-75
Generating Functions....Pages 77-84
The Integral....Pages 85-99
Moments and Laws of Large Numbers....Pages 101-130
Convergence Theorems....Pages 131-143
Conditional Expectations....Pages 145-168
Martingales....Pages 169-188
Optional Sampling Theorems....Pages 189-203
Martingale Convergence Theorems and Their Applications....Pages 205-215
Backwards Martingales and Exchangeability....Pages 217-230
Convergence of Measures....Pages 231-243
Probability Measures on Product Spaces....Pages 245-271
Characteristic Functions and the Central Limit Theorem....Pages 273-293
Infinitely Divisible Distributions....Pages 295-330
Markov Chains....Pages 331-349
Convergence of Markov Chains....Pages 351-388
Markov Chains and Electrical Networks....Pages 389-410
Ergodic Theory....Pages 411-438
Brownian Motion....Pages 439-456
Law of the Iterated Logarithm....Pages 457-508
Large Deviations....Pages 509-519
The Poisson Point Process....Pages 521-541
The Itô Integral....Pages 543-561
Stochastic Differential Equations....Pages 563-588
Back Matter....Pages 589-611
....Pages 613-638