Why is $X_1 + X_2 +\ldots + X_n$ a martingale?

Question

If we have $X_k$ random variables with average $0$ and independent, why is the $\sum_{k=1}^n X_k$ a martingale for the sigma algebra $\mathcal F_n$ generated by $\{X_1,\ldots, X_n\}$?

I basically only have to prove that the expected value of $X_{n+1}$ knowing $\mathcal F_n$ is $0$, but somehow this isn't intuitive to me at all. Could anyone give me pointers on this?

Answer 1

Just check the definition. Let $S_n:=\sum_{k=1}^nX_k$.

• We have that $S_n$ is integrable since so are $X_j$.
• Let $n$ an integer. We have by linearity of conditional expectation that $$E[S_{n+1}\mid\mathcal F_n]=E[S_n+X_{n+1}\mid\mathcal F_n]=E[S_n\mid\mathcal F_n]+ E[X_{n+1}\mid\mathcal F_n].$$ Since $S_n$ is $\mathcal F_n$ measurable, we have $E[S_n\mid\mathcal F_n]=S_n$ and since $X_{n+1}$ is independent of $\mathcal F_n$, we have that $E[X_{n+1}\mid \mathcal F_n]=E[X_{n+1}]=0$.