Hewitt-Savage Theorem

This final instalment in my exploration of exchangeability gives a stronger version of Kolmogorov’s 0-1 law, and suggests some applications. It is easy to see that the tail sigma-field is a subset of the exchangeable sigma-field. For, if A is a tail event, then it is independent of the first n random variables in the underlying sequence, so in particular, is invariant under permutations of initial segments of the sequence.

Kolmogorov’s 0-1 Law: (X_n) a sequence of independent (not necessarily iid) random variables in some probability space. Define the tail sigma-field \tau=\cap_n \sigma(X_{n+1},X_{n+2},\ldots). Then \tau is trivial; that is, \forall A\in\tau\; P(A)\in\{0,1\}.

Proof: Set \tau_n=\sigma(X_{n+1},X_{n+2},\ldots), F_n=\sigma(X_1,\ldots,X_n). Then F_n is independent of \tau_m whenever $m\geq n$. So F_n is independent of \tau for all n, hence so is \cup_n F_n, which generates the entire sigma-field F_\infty, so this is independent of \tau also. Since A\in \tau\Rightarrow A\in F_\infty trivially, the independence criterion gives P(A)=P(A\cap A)=P(A)P(A), and hence P(A)\in\{0,1\}.

Hewitt-Savage 0-1 Law: (X_n) a sequence of iid random variables. Then the sigma field of exchangeable events \mathcal{E} is trivial.

Proof: Take A\in\mathcal{E}, and approximate by A_n\in F_n, P(A\triangle A_n)\rightarrow 0 which is possible, since \cup F_n generates the whole sigma-field. Write A_n=\{(X_1,\ldots,X_n)\in B_n\} for later ease of notation. To exploit exchangeability, set \tilde{A}_n=\{X_{n+1},\ldots,X_{2n}\in B_n\}, as the permutation of RVs that sends A_n\mapsto \tilde{A}_n leaves A invariant. So P(\tilde{A}_n\triangle A)=P(A_n\triangle A)\rightarrow 0\Rightarrow P(A_n\cap \tilde{A}_n)\rightarrow P(A). But because (X) is iid (Note, this is where we use identical distributions), P(A_n\cap \tilde{A}_n)=P(A_n)P(\tilde{A}_n)=P(A_n)^2\rightarrow P(A)^2. Hence P(A)\in\{0,1\}.

Application: Given a stochastic process with iid increments, the event that a state is visited infinitely often is in the tail space of the process, however it is not in the tail space of the increments, so Kolmogorov does not apply. It is however an exchangeable event, and so occurs with probability 0 or 1.

References (for this and the related two previous posts):

Kingman – Uses of Exchangeability (1978)

Breiman – Probability, Chapter 3

Zitkovic – Theory of Probability Lecture Notes

 

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