$E(X_n)=E(X_{n-1})=\ldots=E(X_0)\nRightarrow (X_n)_n$ is Martingale

Question

I am searching for an example of an adapted process $(X_n)_n$ with constant expected value which is not an martingale (I know that the reverse direction holds)

Answer 1

Take $(X_n)_{n\geqslant 1}$ an i.i.d. sequence where $X_1$ is a non-degenerated zero-mean random variable and $\mathcal F_n :=\sigma(X_1,\dots,X_n)$ for $n\geqslant 1 $. Then $\mathbb E[X_n\mid\mathcal F_{n-1}]=0$ for each $n$.