I have a question which describes a system where successive samples are drawn from a Gaussian distribution, and each sample is defined with the following relation.
$v_{k+1} = \gamma v_k + \eta_k$
where $\eta_k \sim \mathcal{N}(0, \sigma^2)$ and $v_0 \sim \mathcal{N}(0, \sigma_v^2)$. $\gamma$ is some positive constant.
Now, I have already figured out the joint probability distribution $P(v_k, v_{k+1})$, and I also know the probability distribution over $P(v_{k+1})$ is simply the marginal probability of $P(v_{k+1})$ derived from the joint probability of $P(v_k, v_{k+1})$.
The part of the question I am facing trouble with is where I need to find out the $\sigma_v$ such that the distribution of $v_k$ is independent of $k$. How would I go about approaching this?
EDIT: Is this an example of an AR(1) model?