How to show these two random variables are orthogonal in $L^2$?

Question

Let $\mathcal{F}_n$ be a decreasing sequence of sub-sigma algebra with $\mathcal{F}=\mathcal{F}_0.$ Let $X$ be a random variable that is in $L^2.$ Let $$M_n= E[X|\mathcal{F}_{n}]-E[X|\mathcal{F}_{n+1}],$$ then I want to show that $E[M_nM_k] = 0,$ where $k<n.$ For starters we want to simplify the following product: \begin{align*} E[(E[X|\mathcal{F}_{n}]-E[X|\mathcal{F}_{n+1}])(E[X|\mathcal{F}_{k}]-E[X|\mathcal{F}_{k+1}])] \end{align*} I think that $$E[X|\mathcal{F}_n]E[X|\mathcal{F}_k] = E[X|\mathcal{F}_k]^2, \quad E[X|\mathcal{F}_{n+1}]E[X|\mathcal{F}_{k+1}]=E[X|\mathcal{F}_{k+1}]^2$$ and that $$E[X|\mathcal{F}_{n+1}]E[X|\mathcal{F}_k] = E[X|\mathcal{F}_k]^2,\quad E[X|\mathcal{F}_n]E[X|\mathcal{F}_{k+1}]=E[X|\mathcal{F}_{k+1}]^2$$ and therefore by adjusting the signs we get that the expectation is $0.$ But I am not sure how to prove the identities for the each product. Any rigorous argument would be much appreciated.

Answer 1

You identity is not correct.

Let $k<n$, that means $$\mathcal{F}_n \subset\mathcal{F}_k$$ because the filtration $\mathcal{F}_n$ is decreasing.

$$E[M_nM_k]=E[E[M_nM_k|\mathcal{F}_n]]=E[M_nE[M_k|\mathcal{F}_n]]$$

We calculate $E[M_k|\mathcal{F}_n]$,

$$E[M_k|\mathcal{F}_n]=E[E[X|\mathcal{F}_k]|\mathcal{F}_n]-E[E[X|\mathcal{F}_{k+1}]|\mathcal{F}_n]$$

We have $$E[E[X|\mathcal{F}_k]|\mathcal{F}_n]=E[X|\mathcal{F}_n]$$ $$E[E[X|\mathcal{F}_{k+1}]|\mathcal{F}_n]=E[X|\mathcal{F}_n]$$ because $$\mathcal{F}_n \subset\mathcal{F}_k$$ $$\mathcal{F}_n \subset\mathcal{F}_{k+1}$$

or $$\mathcal{F}_n \subset\mathcal{F}_k$$ $$\mathcal{F}_n =\mathcal{F}_{k+1}$$ therefore $$E[M_k|\mathcal{F}_n]=0$$

Finally, $$E[M_nM_k]=0$$