Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 201 / 3 / b

Let ${\mathcal{G}}_n=\sigma (S_n,S_{n+1},\dots )$. Since $X_{n+j}=S_{n+j}-S_{n+j-1}$,Permuting $X_1,\dots ,X_n$ preserves their joint probability distribution and leaves every generator of ${\mathcal{G}}_n$ unchanged. Therefore, for each bounded ${\mathcal{G}}_n$-measurable $H$,This is the symmetry of exchangeable random variables. By the defining identity of conditional expectation, all $\mathbb{E}[X_j\mid {\mathcal{G}}_n]$ are equal almost surely. Summing these conditional expectations and using that $S_n$ is ${\mathcal{G}}_n$-measurable givesHence the conditional expectation of a summand given future partial sums isThe integrability of each $X_j$ justifies every conditional expectation above.