"Markov chain {Xn} is inreducible and positive reccurent, the transimission function $P_{ij}(t)$ is differentiable about t for any state i,j.
prove the stationary distribution $\pi$ of {Xn} is also differentiable about t."
I know the $\pi(j)= lim_{n\rightarrow\infty}P_{ij}^{n}$ for aperiodic X.
$\pi(j)= lim_{n\rightarrow\infty}(1/d*\sum_{s=1}^{d}P_{ij}^{nd+s})$ for periodic X with period T.
but how to prove next?