The Annals of Probability

Finite Exchangeable Sequences

P. Diaconis and D. Freedman

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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 $2 ck/n$.

Article information

Ann. Probab., Volume 8, Number 4 (1980), 745-764.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G10: Stationary processes
Secondary: 60J05: Discrete-time Markov processes on general state spaces

de Finetti's theorem exchangeable symmetric extreme points presentable representable variation distance sampling with and without replacement


Diaconis, P.; Freedman, D. Finite Exchangeable Sequences. Ann. Probab. 8 (1980), no. 4, 745--764. doi:10.1214/aop/1176994663.

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