Definition: Let $(F_n)_{n=1}^{\infty}$ be a filtration on the probability space $(\Omega,F, P)$. A discrete time martingale on $(\Omega,F, P,(F_n)_{n=1}^{\infty})$ is a sequence $(X_n)_{n=1}^{\infty}$ of integrable RVs on $(\Omega,F, P)$ adapted to the filtration $(F_n)_{n=1}^{\infty}$ such that $E[X_{n+1}|F_n]=X_n$
I was just wondering if there were any restrictions on whether or not the sequence $(X_n)_{n=1}^{\infty}$ has to have the same distribution. I'm assuming since they are defined on the same probability space that means that they are identically distributed? However, I'm not completely sure if that assumption is correct.
No, they definitely don't have to be identically distributed. As an example, if $(X_n)_{n=1}^\infty$ is simple random walk on $\mathbb{Z}_n$, then every $X_n$ has a different distribution, but the sequence forms a martingale with respect to $\mathcal{F}_n=\sigma(X_1,\dots,X_n)$.