# How can I show that this is a martingale?

Let $\{X_n:n \geq 1\}$ be a sequence of random variables. Let $S_n = \sum_{i=1}^{n} (X_i-E(X_i|X_1,...,X_{i-1})).$ Show that $\{S_n\}$ is a martingale.

This is what I have so far:

To show it's martingale then : $E(S_{n+1}|F_n)$

=$E(S_{n+1}|S_1,S_2,..,S_n) \rightarrow E(S_{n+1}-S_n|X_1,X_2,..,X_n) =$

$\rightarrow E(X_{n+1}-E(X_n+1|X_1,...,X_n])$ am I going in the right path?

I will write $\mathcal{F}_n = \sigma(X_1,...,X_n)$. Then you're given that $S_n = \sum_{i=1}^{n} X_i-E(X_i \mid \mathcal{F}_{i-1})$. We now have
• Yeah, I knew that the $E(X_i)=X_i$ but I didn't know how to approach the rest thank you. – Killercamin May 7 '17 at 21:54
• It's not true that $E(X_i)=X_i$. That would mean that each $X_i$ is constant. – grndl May 7 '17 at 21:56