I am a bit confused in my reading of martingales. I am using Breiman's book and there is an example that doesn't quite make sense to me. Let us define a sequence of random variables in this way:
Let $Y_0=0$ and $Y_1,Y_2,\ldots$ i.i.d. with distribution $P[Y_i=1]=p$ and $P[Y_i=-1]=1-p$ for some $p\in(0,1)$ . Define $X_n:=\sum_{i=0}^n Y_i$. Then he says: If $p=\frac{1}{2}$ , then $(X_n)_{n\in\mathbb{N}}$ is a martingale with respect to its natural filtration $\mathcal{F}_n^X:=\sigma(X_0,\ldots,X_n)$. I dont really understand this statement, because I have to show that $E[X_{n}\mid\mathcal{F}_{s}]=X_{s}$ whenever $n\geq s$ , and I have no idea why this has to be true. By definition of conditional expectation, for any $A\in\mathcal{F}_{s}$ , we have $\int_{A}E[X_{n}\mid\mathcal{F}_{s}]=\int_{A}X_{n}\, dP=\int_{A}\, X_{s}\, dP$ . For some reason, $p=\frac{1}{2}$ should play a role here. I don't really see it.