I am looking into generalizing the Ornstein-Uhlenbeck process to non-Gaussian noise sources. In the code, I discretized the OU process (Euler-Maryuama discretization) and converted it into an AR(1) model, similar to the answer here, where we end up with
$x_{k+1} = \theta(\mu - x_k)\Delta t + \sigma \varepsilon_k\sqrt{\Delta t}$
Since $\varepsilon_k$ is normally distributed, can I not just replace this with another distribution and get a non-Gaussian OU process? It surely is not that simple?