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## De Finetti’s theorem, the pointwise ergodic theorem, and reverse martingales

De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is invariant under exchanging finitely many coordinates (a symmetric measure), then there is some probability measure $\eta$ on probability measures such that $\mu = \int \nu^\infty \, d \eta(\nu)$.

Further, I know the following.

1. The product measures of the form $\nu^{\infty}$ are the extreme points for the convex set of symmetric measures. They are also ergodic with respect to the group of transformations which exchange finitely many coordinates. So $\mu = \int \nu^\infty \, d \eta(\nu)$ is an ergodic decomposition.

2. For $\mu$-a.e. $x=\{x_i\}_{i\in\mathbb{N}}\in \mathbb{R}^\infty$, there is some probability measure $\nu_x$ on $\mathbb{R}$ such that for all measurable sets open balls $A \subseteq \mathbb{R}$,

(A) $\quad$ ${\displaystyle \lim_{k\rightarrow\infty} \frac{1}{k} \sum_{i<k} \mathbf{1}_A(x_i) = \nu_x(A) }$.

Moreover, if $P^n_k$ is the set of all injective functions $\pi \colon [n] \rightarrow [k]$, then for all bounded continuous functions $f\colon \mathbb{R}^n \rightarrow \mathbb{R}$,

(B) $\quad$ ${\displaystyle \lim_{k\rightarrow\infty} \frac{1}{|P^n_k|} \sum_{\pi \in P^n_k} f(x_{\pi(0)},\ldots ,x_{\pi(n-1)}) = \int_{\mathbb{R}^n} f\, d \nu_x^n}$.

3. Equations (A) and (B) and de Finetti's theorem can all be proved using reverse martingales. Indeed, $M_{-k}(x) = \frac{1}{|P^n_k|} \sum_{\pi \in P^n_k} f(x_{\pi(0)},\ldots ,x_{\pi(n-1)})$ is a reverse martingale.

My questions are as follows.

• To what extent are equations (A) and (B) instances of some variant of the pointwise ergodic theorem? (I guess (A) is just Birkoff's pointwise ergodic theorem with the shift map---although I am not sure why the shift map comes in. But (B) is not so clear to me.)

• When may an ergodic average be represented as a reverse martingale?

• Similarly, for which types of pointwise ergodic theorems and ergodic decompositions is there a proof using reverse martingales?

Pointers to any relevant references would be helpful.

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Typo: "...instances of the some pointwise..." – Quinn Culver Apr 10 2012 at 19:26
It seems there's something wrong with the order of the quantifiers in 2. For instance, if the marginal distribution of $X_1$ is non-atomic, then the limit in (A) won't exist for all $A$ (if $A$ is allowed to depend on $x$, then $A$ could for example pick out certain individual values of the $x_i$ but not others in an inconvenient way). Do you mean: for all $A$, then for $\mu$-a.e. $x$, ...? – James Martin Apr 10 2012 at 20:08
@James Martin: I think it is correct. This isn't true for all x, but for a.e. x. In other words, there are $x$'s so random that the $x_i$ are "nicely" distributed in a way that gives the measure $\nu_x$. For example, if the $x_i$ were uniformly distributed, then (A) would hold for the $\nu_x$ equal to the Lebesgue measure. This is similar to the concept of being a generic in ergodic theory (see terrytao.wordpress.com/2008/02/04/…). Let me know if you still disagree. – Jason Rute Apr 10 2012 at 20:29
@Quinn Culver: Fixed. Thanks! – Jason Rute Apr 10 2012 at 20:31
@James Martin: Ok, I see your point. I could take $A={x_0,x_1, \ldots}$. I got this from a paper, which in turn, got it from Kallenberg, Probabilistic symmetries and invariance principles, Proposition 1.4: books.google.com/… I must be reading the a.s. in that statement incorrectly. I think it works if I assume $A$ ranges over all open balls. (I could also use continuous functions like equation (B)). – Jason Rute Apr 10 2012 at 21:40
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It was recently pointed out by Bill Johnson in a comment to a question concerning "Fully exchangeable random sequences" that if an infinite sequence of random variables is exchangeable, then it is "fully exchangeable", that is to say its distribution is in fact invariant by any permutation (even if the permutation has no fixed point). The same argument shows that the distribution of an infinite exchangeable sequence of random variables is invariant under the shift map. Hence you can view (A) as the pointwise ergodic theorem for the shift map.

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