This book covers the fundamentals of measure theory and probability theory. It begins with the construction of Lebesgue measure via Caratheodory’s outer measure approach and goes on to discuss integration and standard convergence theorems and contains an entire chapter devoted to complex measures, Lp spaces, Radon–Nikodym theorem, and the Riesz representation theorem. It presents the elements of probability theory, the law of large numbers, and central limit theorem. The book then discusses discrete time Markov chains, stationary distributions and limit theorems. The appendix covers many basic topics such as metric spaces, topological spaces and the Stone–Weierstrass theorem.
Author(s): Siva Athreya, V. S. Sunder
Publisher: CRC Press
Year: 2008
Language: English
Pages: 118
Tags: Statistics, Probability, Measure Theory, Markov Chains
Probabilities and Measures
Introduction
σ-algebras as events
Algebras, monotone classes, etc.
Preliminaries on measures
Outer measures and Caratheodory extension
Lebesgue measure
Regularity
Bernoulli trials
Integration
Measurable functions
Integration
a.e. considerations
Random Variables
Distribution and expectation
Independent events and tail σ-algebra
Some distributions
Conditional expectation
Probability Measures on Product Spaces
Product measures
Joint distribution and independence
Probability measures on infinite product spaces
Kolmogorov consistency theorem
Characteristics and Convergences
Characteristic functions
Modes of convergence
Central limit theorem
Law of large numbers
Markov Chains
Discrete time MC
Examples
Classification of states
Strong Markov property
Stationary distribution
Limit theorems
Some Analysis
Complex measures
Lp spaces
Radon–Nikodym theorem
Change of variables
Differentiation
The Riesz representation theorem
Appendix
Metric spaces
Topological spaces
Compactness
The Stone–Weierstrass theorem
Tables
References
Index
