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August, 1980 Finite Exchangeable Sequences
P. Diaconis, D. Freedman
Ann. Probab. 8(4): 745-764 (August, 1980). DOI: 10.1214/aop/1176994663


Let $X_1, X_2,\cdots, X_k, X_{k+1},\cdots, X_n$ be exchangeable random variables taking values in the set $S$. The variation distance between the distribution of $X_1, X_2,\cdots, X_k$ and the closest mixture of independent, identically distributed random variables is shown to be at most $2 ck/n$, where $c$ is the cardinality of $S$. If $c$ is infinite, the bound $k(k - 1)/n$ is obtained. These results imply the most general known forms of de Finetti's theorem. Examples are given to show that the rates $k/n$ and $k(k - 1)/n$ cannot be improved. The main tool is a bound on the variation distance between sampling with and without replacement. For instance, suppose an urn contains $n$ balls, each marked with some element of the set $S$, whose cardinality $c$ is finite. Now $k$ draws are made at random from this urn, either with or without replacement. This generates two probability distributions on the set of $k$-tuples, and the variation distance between them is at most $2 ck/n$.


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P. Diaconis. D. Freedman. "Finite Exchangeable Sequences." Ann. Probab. 8 (4) 745 - 764, August, 1980.


Published: August, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0434.60034
MathSciNet: MR577313
Digital Object Identifier: 10.1214/aop/1176994663

Primary: 60G10
Secondary: 60J05

Keywords: De Finetti's theorem , Exchangeable , extreme points , presentable , representable , sampling with and without replacement , Symmetric , variation distance

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 4 • August, 1980
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