Exchangeability and related topics.

*(English)*Zbl 0562.60042
École d’été de probabilités de Saint-Flour XIII - 1983, Lect. Notes Math. 1117, 1-198 (1985).

[For the entire collection see Zbl 0549.00022.]

A sequence \(\{Z_ n,n\geq 1\}\) of random variables is called exchangeable (or interchangeable) if the vectors \((Z_ 1,Z_ 2,...)\) and \((Z_{\pi (1)},Z_{\pi (2)},...)\) are identically distributed for every permutation \(\pi\) such that max\(\{\) \(n: \pi\) (n)\(\neq n\}<\infty\). The result of de Finetti that each infinite exchangeable sequence is a mixture of independent, identically distributed sequences has been known for decades, but significant interest in the theory of exchangeable sequences has only arisen in the past ten or fifteen years. The monograph under review appears to be the first publication to gather the results about exchangeability and to present a unified, detailed treatment of the topic.

The monograph is divided into four parts. The first part discusses directing random measures, mixtures of i.i.d. random variables and their connections with exchangeability; de Finetti’s theorem is proved here. Extensions of exchangeability are the focus of part II; these include sequences of exchangeable pairs, continuous-time processes with exchangeable increments, extensions of de Finetti’s theorem to S-valued random variables and random partitions, and the connection between exchangeability and Dacunha-Castelle’s ”spreading invariance” property and Kallenberg’s stopping time property. Because exchangeable sequences are equivalent to classes of distributions which are invariant under certain transformations, part III is devoted to a study of such transformations, and leads to the notions of partial and weak exchangeability. The last part treats some topics of current research interest, such as exchangeable random sets, sufficiency, population genetics, and urn processes. A list of open problems, an appendix on conditional independence and an extensive bibliography are also provided.

Any researcher in probability theory who would like to update his or her knowledge about exchangeability would be well-advised to read this monograph. The material is complete, well-organized and presented in a careful, readable fashion.

A sequence \(\{Z_ n,n\geq 1\}\) of random variables is called exchangeable (or interchangeable) if the vectors \((Z_ 1,Z_ 2,...)\) and \((Z_{\pi (1)},Z_{\pi (2)},...)\) are identically distributed for every permutation \(\pi\) such that max\(\{\) \(n: \pi\) (n)\(\neq n\}<\infty\). The result of de Finetti that each infinite exchangeable sequence is a mixture of independent, identically distributed sequences has been known for decades, but significant interest in the theory of exchangeable sequences has only arisen in the past ten or fifteen years. The monograph under review appears to be the first publication to gather the results about exchangeability and to present a unified, detailed treatment of the topic.

The monograph is divided into four parts. The first part discusses directing random measures, mixtures of i.i.d. random variables and their connections with exchangeability; de Finetti’s theorem is proved here. Extensions of exchangeability are the focus of part II; these include sequences of exchangeable pairs, continuous-time processes with exchangeable increments, extensions of de Finetti’s theorem to S-valued random variables and random partitions, and the connection between exchangeability and Dacunha-Castelle’s ”spreading invariance” property and Kallenberg’s stopping time property. Because exchangeable sequences are equivalent to classes of distributions which are invariant under certain transformations, part III is devoted to a study of such transformations, and leads to the notions of partial and weak exchangeability. The last part treats some topics of current research interest, such as exchangeable random sets, sufficiency, population genetics, and urn processes. A list of open problems, an appendix on conditional independence and an extensive bibliography are also provided.

Any researcher in probability theory who would like to update his or her knowledge about exchangeability would be well-advised to read this monograph. The material is complete, well-organized and presented in a careful, readable fashion.

Reviewer: R.J.Tomkins

##### MSC:

60G09 | Exchangeability for stochastic processes |

60G07 | General theory of stochastic processes |

60E05 | Probability distributions: general theory |

60G57 | Random measures |

60B11 | Probability theory on linear topological spaces |