Ebook: Probability Theory: A Comprehensive Course

Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.

This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book 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 processes and infinite divisibility;
• large deviation principles and statistical physics;
• Brownian motion; and
• 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 the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.

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Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.

This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book 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 processes and infinite divisibility;
• large deviation principles and statistical physics;
• Brownian motion; and
• 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 the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.

Content:
Front Matter....Pages I-XII
Basic Measure Theory....Pages 1-48
Independence....Pages 49-75
Generating Functions....Pages 77-84
The Integral....Pages 85-99
Moments and Laws of Large Numbers....Pages 101-127
Convergence Theorems....Pages 129-142
Conditional Expectations....Pages 143-167
Martingales....Pages 169-187
Optional Sampling Theorems....Pages 189-203
Martingale Convergence Theorems and Their Applications....Pages 205-215
Backwards Martingales and Exchangeability....Pages 217-229
Convergence of Measures....Pages 231-243
Probability Measures on Product Spaces....Pages 245-270
Characteristic Functions and the Central Limit Theorem....Pages 271-292
Infinitely Divisible Distributions....Pages 293-326
Markov Chains....Pages 327-343
Convergence of Markov Chains....Pages 345-378
Markov Chains and Electrical Networks....Pages 379-402
Ergodic Theory....Pages 403-430
Brownian Motion....Pages 431-445
Law of the Iterated Logarithm....Pages 447-494
Large Deviations....Pages 495-504
The Poisson Point Process....Pages 505-524
The It? Integral....Pages 525-542
Stochastic Differential Equations....Pages 543-566
Back Matter....Pages 567-590
....Pages 591-621