In the proof of De Finetti’s Theorem in my last post, I got to a section where I needed to show a particular convergence property of a sequence of exchangeable random variables. For independent identically distributed RVs, we have Kolmogorov’s 0-1 law, and in particular a strong law of large numbers. Does a version of this result hold for exchangeable sequences? As these represent only a mild generalisation of iid sequences, we might hope so. The following argument demonstrates that this is true, as well as providing a natural general proof of De Finetti.
Define , the smallest sigma-field wrt which the first n RVs are exchangeable. Note that , the exchangeable sigma-field.
So now take g(X) symmetric in the first n variables. By exchangeability . Now set , for , and so because the LHS integrand is -meas. we have . So Z is a backwards martingale.
We have a convergence theorem for backwards martingales, which tells us that exists, and in fact almost surely. Setting gives that . We now perform a similar procedure for functions defined on the first k RVs, in an attempt to demonstrate independence.
For , we seek a backwards martingale, so we take sums over the ways to choose k of the first n RVs. So is a backwards martingale, and hence . As before, set . Crucially, we can replace the falling factorial term with as we are only considering the limit, then exchange summation as everything is positive and nice to get: thus demonstrating independence of conditional on .
So what have we done? Well, we’ve certainly proven de Finetti in the most general case, and we have in addition demonstrated the existence of a Strong Law of Large Numbers for exchangeable sequences, where the limit variable is -measurable.