# A square integrable martingale has orthogonal increments

I am really stuck with the following exercise:

$(\Omega,\mathcal{F},(\mathcal{F}_n)_{n\ge1},P)$ a filtered probability space. Let $(X_n )_{n\ge1}$ be a sequence of square-integrable random variables. Define for every $n\ge1$, $S_n := X_1 +\dots + X_n$ . Suppose that $(X_n )_{n\geq1}$ is such that $(S_n )_{n\geq1}$ is a martingale. Show that if $i \neq j$, then $E[X_i X_j ] = 0.$

So, from the definition of a martingale, we know that $E[S_{n+1}|\mathcal{F}_n]=S_n, \forall n\in\mathbb{N}$, but I don't know how to use this to prove the claim.

Thanks for your help!

## 1 Answer

All the conditional expectations make sense since $X_n$ is square integrable for each $n$.

Assume that $i\lt j$. Then $X_iX_j$ has the same expectation as $\mathbb E[X_iX_j\mid\mathcal F_i]$, which is equal to $X_i\mathbb E[X_j\mid\mathcal F_i]$. Since $\mathcal F_i\subset \mathcal F_{j-1}$, we can conclude by the towering property that $E[X_j\mid\mathcal F_i]=0$.

• Thanks! I just don't quite understand why we know that XiXj has the same expectation as E[XiXj|Fi]. – caedmon Oct 5 '13 at 15:24
• I agree with this, although it might be worth mentioning where the square integrability condition is used. – George Lowther Oct 5 '13 at 15:25
• Use the definition of conditional expectation with $\Omega$. – Davide Giraudo Oct 5 '13 at 15:25
• Ok, I get it. And where do we use that (Sn) is a martingale? – caedmon Oct 5 '13 at 15:38
• In the last equality ($\mathbb E[X_j\mid\mathcal F_i]=0$). – Davide Giraudo Oct 5 '13 at 15:39