Exchangeable random variables

An exchangeable sequence of random variables is a sequence X₁, X₂, X₃, ... 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

${\displaystyle X_{\sigma (1)},X_{\sigma (2)},X_{\sigma (3)},\dots }$

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

A sequence E₁, E₂, E₃, ... 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_X1,...,Xn(x₁, ... ,x_n) of a finite sequence of exchangeable random variables is symmetric in its arguments x₁, ... ,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 ith marble drawn is red. Then {X_i}_i=1,...n is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.

Properties

• Covariance: for a finite exchangeable sequence { X_i }_{i = 1, 2, 3, ...} of length n:

${\displaystyle \operatorname {cov} (X_{i},X_{j})={\text{constant}}\geq {\frac {-\sigma ^{2}}{n-1}},\quad {\text{for }}i\neq j,}$ 

where σ ² = var(X₁).

"Constant" in this case means not depending on the values of the indices i and j as long as i ≠ j.

This may be seen as follows:

${\displaystyle {\begin{aligned}0&\leq \operatorname {var} (X_{1}+\cdots +X_{n})=\operatorname {var} (X_{1})+\cdots +\operatorname {var} (X_{n})+\underbrace {\operatorname {cov} (X_{1},X_{2})+\cdots } _{\text{all ordered pairs}}\\&=n\sigma ^{2}+n(n-1)\operatorname {cov} (X_{1},X_{2}),\end{aligned}}}$ 

and then solve the inequality for the covariance. Equality is achieved in a simple urn model: An urn contains 1 red marble and n − 1 green marbles, and these are sampled without replacement until the urn is empty. Let X_i = 1 if the red marble is drawn on the ith trial and 0 otherwise.

the case in which Xi is

• For an infinite exchangeable sequence,

${\displaystyle \operatorname {cov} (X_{i},X_{j})={\text{constant}}\geq 0.\,}$ 

See also

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

References

• Aldous, David J., Exchangeability and related topics, in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp. 1-198, Springer, Berlin, 1985. ISBN 978-3-540-15203-3 DOI 10.1007/BFb0099421
• 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