$\dfrac{1}{T}\underset{t = 0}{\overset{T - 1}{\sum}}xP^t$ converges

Question

I'm trying to proof that $\dfrac{1}{T}\underset{t = 0}{\overset{T - 1}{\sum}}xP^t$, where x is stochastic vector, P is stochastic matrix, converges. I understand that if there are no eigenvalues of $\pi = \dfrac{1}{T}\underset{t = 0}{\overset{T - 1}{\sum}}xP^t$ such that $|\lambda| = 1, \; \lambda \neq 1,$ $\pi$ converges. But I can't understand what if $P$ has eigenvalues $\lambda \neq 1,$ such that $|\lambda| = 1$. Thank you for any help!