By combining two methods I've stumbled into a rather messy random walk situation. I have the typical random walk setup


$$\theta\_{i+1} = \theta\_{i} + \hat{\theta}\_{i+1}$$


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. 



Is this a random walk? Does it have a name?
I presume you mean $\epsilon = \epsilon\_i$, where $\epsilon\_i \sim \mathscr N(0,1)$ is independent of all the previous random variables. The pairs $(\theta\_i, \hat{\theta}\_i)$ form a Gaussian Markov process.


