This book is about random objects-sequences, processes, arrays, measures, functionals-with interesting symmetry properties. Here symmetry should beunderstoodinthebroadsenseofinvarianceunderafamily(notnecessarily a group) of measurable transformations. To be precise, it is not the random objects themselves but rather their distributions that are assumed to be symmetric. Though many probabilistic symmetries are conceivable and have been considered in various contexts, four of them-stationarity, contractability, exchangeability, and rotatability-stand out as especially interesting and - portant in several ways: Their study leads to some deep structural theorems of great beauty and signi?cance, they are intimately related to some basic areasofmodernprobabilitytheory, andtheyaremutuallyconnectedthrough a variety of basic relationships. The mentioned symmetries may be de?ned as invariance in distribution under shifts, contractions, permutations, and rotations. Stationarity being a familiar classical topic, treated extensively in many standard textbooks and monographs, most of our attention will be focused on the remaining three basic symmetries. The study of general probabilistic symmetries essentially originated with the work of de Finetti (1929-30), who proved by elementary means (no - vanced tools being yet available) the celebrated theorem named after him- the fact that every in?nite sequence of exchangeable events is mixed i.i.d.

About the Author

Olav Kallenberg received his Ph.D. in 1972 from Chalmers University in Gothenburg, Sweden. After teaching for many years at Swedish universities, he moved in 1985 to the US, where he is currently Professor of Mathematics at Auburn University. He is well known for his previous books Random Measures (4th edition, 1986) and Foundations of Modern Probability (2nd edition, 2002) and for numerous research papers in all areas of probability. In 1977, he was the second recipient ever of the prestigious Rollo Davidson Prize from Cambridge University. In 1991-94, he served as the Editor in Chief of Probability Theory and Related Fields. Professor Kallenberg is an elected fellow of the Institute of Mathematical Statistics.

Reviews

From the reviews of the first edition:

"The starting point of this book ... is de Finetti's famous theorem characterizing sequences of exchangeable random variables as mixtures of i.i.d. random variables. ... The monograph provides a tight approach to this subject based on modern mathematical tools ... and presents de Finetti's theorem in a new framework ... . the monograph under review presents ... a profound and complete treatment of the topic. The reader interested in this field will find a comprehensive investigation leading to the actual frontiers of research." (Wilfried Hazod, Mathematical Reviews, Issue 2006 i)

"This elegantly written, graduate level monograph provides a comprehensive coverage of probabilistic symmetries. ... This book has an extensive bibliography and valuable, detailed historical notes that place the work in context and provide links to areas of application. The text conveys the author's enthusiasm for the subject and his deep understanding of the key structures gained over 30 years of active research into the subject." (Neville Weber, Zentralblatt MATH, Vol. 1084, 2006)

"The author achieves a remarkable level of clarity, economy, and accessibility ... with enviable skill and judgement. ... The book may thus be used as much as read in the future. ... A reader interested in exchangeability will find results a-plenty in this book ... . As a whole, this is a grand conception, superbly realized." (Charles M. Goldie, Journal of the American Statistical Association, Vol. 102 (479), 2007)

Book Information
ISBN 9781441920423
Author Olav Kallenberg
Format Paperback
Page Count 512
Imprint Springer-Verlag New York Inc.
Publisher Springer-Verlag New York Inc.