exchangeable random variables
A finite set of random variables
{X1,…,Xn} defined on a common probablility space (Ω,ℱ,P) is said to be exchangeable if
P((X1∈B1)∩⋯∩(Xn∈Bn))=P((Xσ(1)∈B1)∩⋯∩(Xσ(n)∈Bn)) |
for every set of Borel sets {B1,…,Bn}, and every permutation σ∈Sn. In other words, X1,…,Xn are exchangeable if their joint probability distribution function is the same regardless of their order.
A stochastic process {Xi} is said to be exchangeable if every finite subset of {Xi} is exchangeable.
Remarks
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•
If S={X1,…,Xn} is exchangeable, then every subset of S is exchangeable (by picking suitable Bi and σ). In particular, all Xi are identically distributed, for
P(Xi∈B)=P((Xi∈B)∩(Xj∈ℝ))=P((Xj∈B)∩(Xi∈ℝ))=P(Xj∈B). -
•
If S={X1,…,Xn} is iid, then S is exchangeable, since the joint distribution
of Xi is the product
of the distributions
of Xi:
P((X1∈B1)∩⋯∩(Xn∈Bn))=P(Xσ(1)∈B1))⋯P(Xσ(n)∈Bn)).
Title | exchangeable random variables |
---|---|
Canonical name | ExchangeableRandomVariables |
Date of creation | 2013-03-22 16:25:53 |
Last modified on | 2013-03-22 16:25:53 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 5 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 60G09 |
Synonym | exchangeable stochastic process |
Defines | exchangeable |
Defines | exchangeable process |