sequence of bounded random variables form martingale then they are uncorrelated

Question

Let $X_i$ be a sequence of bounded random variables such that $S_n=\sum\limits_{i=1}^{n} X_i$ is an martinagle w.r.t the filtration $\mathcal F$. Could anyone tell me how to show $$Cov(X_i, X_j)=0\quad \forall i\ne j$$

Thanks.

Answer 1

Using the fact that $(S_n)$ is a martingale with respect to $(\mathcal{F}_n)$, you can show that $E[X_{n+1} \mid \mathcal{F}_n] = 0$.

So, if $i < j$, one can show that $E[X_j] = E[E[X_j \mid \mathcal{F}_i]] = 0$ and $E[X_i] = E[E[X_i \mid \mathcal{F}_0]] = 0$, so

$$\text{Cov}(X_i ,X_j) = E[X_i X_j] = E[X_i E[X_j \mid \mathcal{F}_i]] = 0.$$