1992 6 X 9 440 pp.
Hardback 0-521-35023-9 or 978-0521-35023-5
U. K. £90.00
U. S. $150.00
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Paperback 978-0521-13508-5 U. K. £29.99 U. S. $55.00 Best Book Buys
A basic notion in the book is that of a "stopping time." The technique of stopping times is used in the context of directed processes and provides many applications in probability and analysis. Martingales and amarts (asymptotic martingales) play an important role.
The book opens with a discussion of pointwise and stochastic convergence of processes with concise proofs arising from the method of stochastic convergence. Later, the rewording of Vitali covering conditions in terms of stopping times clarifies connections with the theory of stochastic processes. Solutions are presented here for nearly all the open problems in the Krickeberg convergence theory for martingales and submartingales indexed by directed set.
Another theme is the unification of martingale and ergodic theorems. Among the topics treated are: the three-function maximal inequality, Burkholder's martingale transform inequality, prophet inequalities, convergence in Banach spaces, and a general superadditive ratio ergodic theorem. From this, the general Chacon-Ornstein theorem and the Chacon theorem can be derived. A second instance of the unity of ergodic and martingale theory is a general principle showing that in both theories, all the multiparameter convergence theorems follow from one-parameter maximal and convergence theorems.
Contents: Stopping times/ Infinite measure and Orlicz spaces/ Inequalities/ Directed index set/ Banach-valued random variables/ Martingales/ Derivation/ Pointwise ergodic theorems/ Multiparameter processes
There is an errata file available:
errata.pdf (94 K) PDF format