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.