Varying definitions of a martingale

Question

I frequently see that, over some filtration $\mathscr{F}_{n}, \{X_{n}\}$ is defined as a martingale if $E[X_{n+1}|\mathscr{F}_{n}]=X_{n}$. Sometimes, however, I see this extended to $E[X_{n+s}|\mathscr{F}_{n}]=X_{n}, S\geq1$. It seems this may follow from the case where $X_{n}$ is Markovian and therefore that this second definition is merely generalizing the first over Markovian and non-Markovian martingales. Is this more or less the case?

Answer 1

The two definitions are just equivalent by the tower law for conditional expectation. Clearly the second implies the first. To see that the first implies the second, write for $S>1$, $$\mathbb{E}[X_{n+s} \mid \mathcal{F}_n] = \mathbb{E}[ \mathbb{E}[X_{n+s} \mid \mathcal{F}_{n+s-1}] \mid \mathcal{F}_{n}] = \mathbb{E}[X_{n+s-1} \mid \mathcal{F}_n]$$ where the last equality follows by applying the first definition. At this point, it should be clear that repeating this argument gives $\mathbb{E}[X_{n+s} \mid \mathcal{F}_n] = X_n$ as in the second definition.