Suppose to consider a generic Markov Chain, suppose $P$ to be the unknown transition matrix and $\pi$ its stationary distribution.
Assume $Var[P]$ to be known (estimated from some data).
(Variance is calculated element-wise)
We know that:
$\pi = \pi P$
I'm interested in $Var[\pi]$, the main problem I'm facing is the lack of independence between $\pi$ and $P$.
Some help? Thank you!