Let's consider a random walk $S_n=\sum_{i=1}^n{X_i}$ starting from the origin, with the following conditions:
finite range, symmetric distribution, irreducibility (with respect to the state space), finite second moment and mean $0$, aperiodicity.
The fact that the distribution is symmetric, implies that the all odd moments are $0$. I would need finite fourth moment. Does it is possible to recover it from these assumptions?