Conditional expectation of a summand given future partial sums
= Conditional expectation of a summand given future partial sums
For <independent and identically distributed random variables> $X_j$ that are <integrable random variables>, with $S_n=\sum_{j=1}^nX_j$, symmetry gives $\mathbb E[X_1\mid S_n,S_{n+1},\ldots]=S_n/n$. The decreasing <sigma-algebras> generated by these future sums permit application of the <reverse martingale convergence theorem>.