I'm facing the following problem:
let $V = \sum_{t=0}^{\infty} P^{t}$
where P is a $n \times n$ matrix.
I want to calculate $Var(V)$, variance is calculated coefficient-wise. I know the variance of each coefficient of $P$ and I also know that this coefficient are correlated with each other.
I've developed a method to approximate this value but it is time expensive. I would like to know if there is a way to obtain a nice closed formula.
Thank you for your help!