If we suppose an instrument goes up or down 1 tick per $\Delta t$ (binary model), its long term distribution will be normal, per the Central Limit Theorem.
However, suppose we model as follows:
The first tick is up or down with 50% probability.
Every future tick is: 60% likely to be in the same direction as the previous tick, 40% likely to be in the opposite direction.
The Central Limit Theorem doesn't apply here, because these are no longer independent random variables.
However, as this post shows us, the resulting distribution is still normal.
My question: can I construct a binary model that yields a non-normal distribution, ideally a "fat tailed" distribution?