Marginal distribution of an autoregressive process of order one AR(1) I'm reading "Econometric Modelling with Time Series" by V. L. Martin, A. S. Hurn and D. Harris ( https://www.researchgate.net/file.PostFileLoader.html?id=56bccdaa6225ff0de28b45a6&assetKey=AS:328005403594753@1455213991422 ).
I don't understand example 1.9 at page 11.
Let's consider an AR(1) model : $Y_t = \rho*Y_{t-1} + \upsilon_t$ $,$
$$\upsilon_t \sim i.i.d.\text{N} \Big( 0, \sigma^{2}\Big),$$ $$|\rho|<1$$
Why is the marginal distribution of $Y_1 \sim \text{N} \Big( 0, \frac{\sigma^2}{1-\rho^2} \Big)$ ?
Thanks for your help!
Here the mentioned example


Consider the $\sf AR(1) $ model: $$ Y_t= c+\rho Y_{t-1}+v_t.\tag 1$$ For $|\rho|< 1,$ the process is covariance stationary and the stable solution is $$Y_t = \sum_{j\in \mathbb N\cup \{0\}}[\rho^j c +\rho^jv_{t-j}].\tag 2\label 2$$ From $\eqref 2,$ it can be easily deduced \begin{align}\mathbb EY_t&=\frac{c}{1-\rho}\tag{3.1},\\\mathbb VY_t&= \frac{\sigma^2}{(1-\rho^2)}\tag{3.2}\end{align}
Since $\langle v_t\rangle$ is Gaussian, $Y_1$ is also Gaussian. Therefore, for $c=0, ~Y_1\sim\mathcal N(0,\sigma^2/(1-\rho^2).$

Reference:
$[\rm I]$ Time Series Analysis, James D. Hamilton, Princeton University Press, $1994, $ sec. $5.2, $ p. $118.$
