Comprising the major theorems of probability theory and the measure theoretical foundations of the subject, the main topics treated here are independence, interchangeability, and martingales. Particular emphasis is placed upon stopping times, both as tools in proving theorems and as objects of interest themselves.
No prior knowledge of measure theory is assumed and a unique feature of the book is the combined presentation of measure and probability. It is easily adapted for graduate students familiar with measure theory using the guidelines given.
Special features include:
- A comprehensive treatment of the law of the iterated logarithm
- The Marcinklewicz-Zygmund inequality, its extension to martingales and applications thereof
- Development and applications of the second moment analogue of Walds equation
- Limit theorems for martingale arrays; the central limit theorem for the interchangeable and martingale cases; moment convergence in the central limit theorem
- Complete discussion, including central limit theorem, of the random casting of r balls into n cells
- Recent martingale inequalities
- Cram r-L vy theorem and factor-closed families of distributions
Contents of Probability Theory
- Classes of Sets, Measures,and Probability Spaces
- Binomial Random Variables
- Independence
- Integration in a Probability Space
- Sums of Independent Random Variables
- Measure Extensions, Lebesgue-Stieltjes Measure, Kolmogorov Consistency Theorem
- Conditional Expectation, Conditional Independence, Introduction to Martingales
- Distribution Functions and Characteristic Functions
- Central Limit Theorems
- Limit Theorems for Independent Random Variables
- Martingales
- Infinitely Divisible Laws
-Index
