*by *
G. A. Edgar & Louis Sucheston

##### (Encyclopedia of Mathematics and Its Applications 47)

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
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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

#### errata

There is an errata file available:

errata.pdf (94 K) PDF format

send comments to
edgar@math.ohio-state.edu