lets say we have a Markov chain with $\pi^T = ¸\pi^T P$, but we do not know anything about $P$ unless that the chains has a unique stationary distribution $p_*$ that it converges to.
Let $e_t = p_t - p_*$ be the error. I would like to know how fast the error converges to zero. So what is $f$ in $\|e_t\|_2 \leq f(t) \|e_0\|_2$
If we assume that $P$ has an Eigendecomposition it"s easy. But how do you do it in the general case?