Exchangeable random variables

Exchangeable random variables

An exchangeable sequence of random variables is asequence "X"1, "X"2, "X"3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., i.e. any permutation σ that leaves all but finitely many indices fixed, the joint probability distribution of the permuted sequence

: X_{sigma(1)}, X_{sigma(2)}, X_{sigma(3)}, dots

is the same as the joint probability distribution of the original sequence.

A sequence "E"1, "E"2, "E"3, ... of events is said to be exchangeble precisely if the sequence of its indicator functions is exchangeable.

Independent and identically distributed random variables are exchangeable.

The distribution function "F""X"1,...,"X""n"("x"1, ... ,"x""n") of a finite sequence of exchangeable random variables is symmetric in its arguments "x"1, ... ,"x""n".

Examples

* Any weighted average of iid sequences of random variables is exchangeble. See in particular de Finetti's theorem.

* Suppose an urn contains "n" red and "m" blue marbles. Suppose marbles are drawn without replacement until the urn is empty. Let "X""i" be the indicator random variable of the event that the "i"th marble drawn is red. Then {"X""i"}"i"=1,..."n" is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.

* Let "X"1, "X"2, "X"3, ... be exchangeable random variables, taking real values and such that E("X""i" 2) < ∞. Then E("X"1 "X"2) ≥ 0.

ee also

* Hewitt-Savage zero-one law
* de Finetti's theorem

References

* Spizzichino, Fabio "Subjective probability models for lifetimes". Monographs on Statistics and Applied Probability, 91. "Chapman & Hall/CRC", Boca Raton, FL, 2001. xx+248 pp. ISBN 1-58488-060-0


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