Exchangeability and related topics
École d'été de probabilités de Saint-Flour, XIII---1983, volume 1117 of Lecture Notes in Math., Springer, Berlin, (1985)Abstract
In some broad sense, exchangeability theory may be regarded as the study of probability distributions on a measurable space $S$ which exhibit some symmetry property, such as invariance under a class of transformations of $S$ (though stationarity forms a subject on its own and is usually not included). The core result is the celebrated theorem of de Finetti, which characterizes infinite exchangeable sequences of random variables (i.e., those for which the distribution is invariant under permutations of the components) as mixtures of i.i.d. sequences. For decades after its discovery in the 1930s, this was essentially the only nontrivial result known in the area (with some notable exceptions). A revival of exchangeability theory occurred in the 1970s, with impetus from both theory and applications, and the field has ever since been in a state of vigorous development.
The present lecture notes form the first modern and comprehensive survey of the area. The author, himself a contributor of some remarkable and deep results in the field, has succeeded in filling every page of his treatise with a stunning insight and richness in ideas, which should make the reading of these notes enjoyable and rewarding for every probabilist, especially since some of the more lengthy and technical proofs have been omitted, and attention is focused instead on principles and basic ideas.
The notes are divided into 21 sections, grouped into four parts. Here Part I gives a modern exposition of de Finetti's theorem,in different versions and with different proofs. Part II starts withRyll-Nardzewski's subsequence characterization of exchangeablesequences and the reviewer's stopping time approach, which leads naturally to a discussion of the author's subsequence principle. This part also contains a brief analysis of exchangeable processes in continuous time, and it ends with a discussion of exchangeable partitions, in the spirit of J. F. C. Kingman\en.
In Part III, the author proceeds to some more complicated structures, involving infinitary trees, random arrays, and infinite-dimen-sional cubes. The central fact for arrays is the characterization of row-column exchangeability, due independently to the author and D. N. Hoover\en. The material on trees and cubes is new, and exhibits some interesting connections to Markov random fields and random walks.
The final part, Part IV, includes miscellaneous topics related to exchangeability, such as random partitions of a line segment, sufficient statistics, sampling from a finite population, and some weak convergence results. There is also a brief account of Kingman's applications of exchangeability to population genetics. One might finally mention the numerous open problems, scattered throughout the notes, which may stimulate a continued rapid growth of the subject.
Description
Exchangeability and related topics
