Is this a random walk? Does it have a name?

Question

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.

Answer 1

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.