# On the existance of finite state Paul-Levy stochastic process .

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?