By combining two methods I've stumbled into a rather messy random walk situation. I have the typical random walk setup
$$\theta_{i+1} = \theta{i} + \hat{\theta}_{i+1}$$$$\theta_{i+1} = \theta_{i} + \hat{\theta}_{i+1}$$
Where $\hat{\theta}_{i+1} \sim \mathcal{N}(0,1)$. However, $\hat{\theta}_{i+1}$ depends on the previous $\hat{\theta}_{i}$:
$$\hat{\theta}_{i+1} = \beta\,\hat{\theta}_{i} + \sqrt{1 - \beta^2}\,\epsilon$$
Where $\beta \in [0,1]$ and $\epsilon \sim \mathcal{N}(0, 1)$.
Does this setup have a name? Does it make sense as a random walk? I have not been able to find examples in the literature, but I am not a statistician.
