The exercise from L. Breiman " Probability " page 76

Question

How to solve this question ?

$\textbf{15.}~$ Let $X_1 , X_2, \dots $ be independent, identically distributed random variables, $E|X_1| < \infty $, and denote $S_n = X_1 + \cdots +X_n.$ Prove that $$ E(X_1 | S_n, S_{n+1}, \dots )= S_n /n ~~~~a.s. $$ [Use symmetry in the final step.]

I have no idea to solve this question. Any hints would be greatly appreciated!

Answer 1

$$S_{n+1}=S_n + X_{n+1}$$

$$S_{n+2}=S_{n+1} + X_{n+2}$$

$$\vdots$$

so that

$$ E(X_1 | S_n, S_{n+1}, \dots )= E(X_1|S_n,X_{n+1},X_{n+2}, \dots) =E(X_1|S_N)$$

By symmetry or exchangability (De Finetti's theorem)

$$E(X_1|S_n) =E(X_j|S_n) , \quad j \in \{1,2,\dots,n\}$$

$$n E(X_1|S_n) =\sum_{j=1}^n E(X_j|S_n) = E(S_n|S_n)=S_n $$

$$ E(X_1 | S_n, S_{n+1}, \dots )=E(X_1|S_n) = \frac{S_n}{n}$$

Answer 2

From definition of conditional expectation you can verify that $E(X_1|S_n,S_{n+1},...)=E(X_i|S_n,S_{n+1},...)$ for $ 1 \leq i \leq n$. Add these $n$ equations and divide by $n$.