First Look at Rigorous Probability Theory

This textbook is an introduction to probability theory using measure theory. It is designed for graduate students in a variety of fields (mathematics, statistics, economics, management, finance, computer science, and engineering) who require a working knowledge of probability theory that is mathematically precise, but without excessive technicalities. The text provides complete proofs of all the essential introductory results. Nevertheless, the treatment is focused and accessible, with the measure theory and mathematical details presented in terms of intuitive probabilistic concepts, rather than as separate, imposing subjects. In this new edition, many exercises and small additional topics have been added and existing ones expanded. The text strikes an appropriate balance, rigorously developing probability theory while avoiding unnecessary detail.

Contents:

• The Need for Measure Theory
• Probability Triples
• Further Probabilistic Foundations
• Expected Values
• Inequalities and Convergence
• Distributions of Random Variables
• Stochastic Processes and Gambling Games
• Discrete Markov Chains
• More Probability Theorems
• Weak Convergence
• Characteristic Functions
• Decomposition of Probability Laws
• Conditional Probability and Expectation
• Martingales
• General Stochastic Processes