Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 28 / 4 / a / Solution

Let $T_{\pi }$ permute coordinates $0,\dots ,m$, fixing the later coordinates. For any integrable random variable $V$, its group averageis invariant under this group, hence ${\mathcal{S}}_m$-measurable. If $C\in {\mathcal{S}}_m$, invariance of $C$ and measure preservation of each $T_{\pi }$ give $\mathbb{E}[1_{C}V\circ T_{\pi }]=\mathbb{E}[1_{C}V]$. Thus $A_{m}V$ satisfies the integral definition of $\mathbb{E}[V\mid {\mathcal{S}}_m]$. This proves symmetrization as conditional expectation.

Apply it to $V=1_{A_k}(X_k)$ for $m\ge k$. Under a uniform permutation of $m+1$ coordinates, its selected coordinate is uniform among all of them, soThe reverse martingale convergence theorem says that, for decreasing sigma-algebras ${\mathcal{H}}_m$ and an integrable $V$, these conditional expectations converge almost surely and in $L^1$ to $\mathbb{E}[V\mid {\bigcap }_m{\mathcal{H}}_m]$. Here ${\mathcal{S}}_m\downarrow \mathcal{S}$, because every finitely supported permutation belongs to one of the finite groups. ThereforeIntersect these probability-one events for the finitely many $k=0,\dots ,p$ and multiply the limits. Expanding the product of finite sums givesThe sum here includes repeated indices. No independence of the original coordinates was assumed; exchangeability and reverse conditional-expectation convergence are precisely what replaced it.