Does there exist a stochastic process $ \{ X_n \}_{n \in N}$ on a probability space $(\Omega, \mathcal F,\operatorname{P}) $ such that:
1) $X_0 = 0$ a.s.
2) $X_n$ has stationary and independent increments
3) $X_n \sim Binomial(n, \frac{1}{2} )$
This point can even be changed in $X_n \sim Multinomial $ .
Ideally I was looking for a stochastic Levy process where fixed a time $n_1$ the random variable $X_{n_1}$ could assume only a finite number of values but as $n$ increased it approximated a discrete time random walk.
Does something of this kind exist?