# Exercise 7.7.1 in Grimmett & Stirzaker's 'Probability and Random Processes'

I'm having trouble solving exercise 7.7.1 in Grimmett & Stirzaker's Probability and Random Processes, which reads:

Let $X_1,X_2,\ldots$ be random variables such that the partial sums $S_n=X_1+X_2+ \cdots + X_n$ determine a martingale. Show that $\mathbb{E}\left(X_iX_j\right)=0$ if $i \neq j$.

To start, I'm just trying to show $\mathbb{E}[X_{1}X_{2}]=0$. I've tried writing \begin{align} X_{1}^{2} &= X_{1} \mathbb{E}[S_{2}|X_{1}] &&\text{(By the martingale property.)}\\ &= \mathbb{E}[X_{1}^{2}|X_{1}] + \mathbb{E}[X_{1}X_{2}|X_{1}] &&\text{(By linearity of conditional expectation.)} \\ &=X_{1}^2 + \mathbb{E}[X_{1} X_{2}|X_{1}] &&\text{(By properties of conditional expectation.)} \end{align} So $\mathbb{E}[X_{1} X_{2}|X_{1}]=0$, though this is not quite what I want.

Am I on the right track? Why or why not?

-

You're practically done, since $\mathbb{E}[X_1 X_2] = \mathbb{E}[\mathbb{E}[X_1 X_2 | X_1]]$.