An Exchangeable Law of Large Numbers

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 \mathcal{E}_n=\sigma(\{f(X_1,\ldots,X_n): f\text{ symmetric, Borel}\}), the smallest sigma-field wrt which the first n RVs are exchangeable. Note that \mathcal{E}_1\supset\mathcal{E}_2\supset\ldots\supset \mathcal{E}=\cap_n\mathcal{E}_n, the exchangeable sigma-field.

So now take g(X) symmetric in the first n variables. By exchangeability E[\frac{1}{n}\sum_1^n f(X_j)g(X)]=E[f(X_1)g(X)]. Now set g=1_A, for A\in\mathcal{E}_n, and so because the LHS integrand is \mathcal{E}_n-meas. we have Z_n=\frac{1}{n}\sum_1^n f(X_j)=E[f(X_1)|\mathcal{E}_n]. So Z is a backwards martingale.

We have a convergence theorem for backwards martingales, which tells us that \lim_n n^{-1}\sum^n f(X_j) exists, and in fact = E[f(X_1)|\mathcal{E}] almost surely. Setting f(X)=1(X\leq x) gives that \lim_n\frac{\#\{X_i\leq x: i\leq n\}}{n}=F(x):=P(X_1\leq x|\mathcal{E}). We now perform a similar procedure for functions defined on the first k RVs, in an attempt to demonstrate independence.

For f:\mathbb{R}^k\rightarrow\mathbb{R}, we seek a backwards martingale, so we take sums over the n^{(k)} ways to choose k of the first n RVs. So \frac{1}{n(n-1)\ldots(n-k+1)}\sum_{I\subset[n]} f(X_{i_1},\ldots,X_{i_k}) is a backwards martingale, and hence E[f(X_1,\ldots,X_k)|\mathcal{E}]=\lim_n \frac{1}{n(n-1)\ldots(n-k+1)}\sum f(-). As before, set f(y_1,\ldots,y_k)=1(y_1\leq x_1)\ldots 1(y_k\leq x_k). Crucially, we can replace the falling factorial term with n^{-k} as we are only considering the limit, then exchange summation as everything is positive and nice to get: E[f(X_1,\ldots,X_k)|\mathcal{E}]=\lim(\frac{1}{n}\sum 1(X_1\leq x_1))\ldots(\frac{1}{n}\sum 1(X_k\leq x_k)) thus demonstrating independence of (X_n) conditional on \mathcal{E}.

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 \mathcal{E}-measurable.

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