The Annals of Probability

Finite Exchangeable Sequences

P. Diaconis and D. Freedman

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Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176994663

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176994663

Mathematical Reviews number (MathSciNet)
MR577313

Zentralblatt MATH identifier
0434.60034

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

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

Citation

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


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