## 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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