I have the following recursive equation,
$\tilde{S}(k+1) = \beta \bar{r}(k) + (1-\beta)\tilde{S}(k)$
where $\bar{r}(k) \sim \mathcal{N}(0, I)$.
How can I calculate the variance of $\tilde{S}(k)$? (about its mean I think it should be zero).
[I know the answer is $\tilde{S}(k) \sim \mathcal{N}(0, \beta\frac{1-(1-\beta)^{2k}}{2-\beta}I)$ but don't know how to reach it.]