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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?

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