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.
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.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 allmeasurable setsopen 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) $\quad$
${\displaystyle \lim_{k\rightarrow\infty} \frac{1}{|P^n_k|} \sum_{\pi \in P^n_k} f(x_{\pi(0)},\ldots ,x_{\pi(n-1)}) = \int_{\mathbb{R}^n} f\, d \nu_x^n}$.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.
$A={x_0,x_1, \ldots}$. 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