Olav Kallenberg - Böcker

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.

This is the first comprehensive treatment of the three basic symmetries of probability theory - contractability, exchangeability, and rotatability - defined as invariance in distribution under contractions, permutations, and rotations. Originating with the pioneering work of de Finetti from the 1930's, the theory has evolved into a unique body of deep, beautiful, and often surprising results, comprising the basic representations and invariance properties in one and several dimensions, and exhibiting some unexpected links between the various symmetries as well as to many other areas of modern probability. Most chapters require only some basic, graduate level probability theory, and should be accessible to any serious researchers and graduate students in probability and statistics. Parts of the book may also be of interest to pure and applied mathematicians in other areas. The exposition is formally self-contained, with detailed references provided for any deeper facts from real analysis or probability used in the book.

Getoor, MetrikaFrom the reviews of the second edition: “This … edition presents … more material in the concise and elegant style of the former edition which by now has become a highly praised standard reference book for many areas of probability theory.” M.

Getoor, MetrikaFrom the reviews of the second edition: “This … edition presents … more material in the concise and elegant style of the former edition which by now has become a highly praised standard reference book for many areas of probability theory.” M.

Offering the first comprehensive treatment of the theory of random measures, this book has a very broad scope, ranging from basic properties of Poisson and related processes to the modern theories of convergence, stationarity, Palm measures, conditioning, and compensation.