Monotonic convergence of powers of a stochastic matrix

Question

Let $P$ be a stochastic matrix (nonnegative and each row summing to 1). Assuming that $P^n$ converges to $\textbf{1}\pi$ as $n \rightarrow \infty$, where $\pi$ is a row vector (stationary distribution of a finite irreducible Markov chain), I am interested in a monotonic (nonovershooting or nonundershooting) convergence of some elements of $P$.

I need this to prove that, starting from some initial high value, the probability of a certain state (or states) in a Markov chain (satisfying some condition) never drops below its stationary value.

I wonder if there is any result discussing the conditions on the elements of $P$ for such a behavior. I am not sure how to tackle this problem. Any help will be really appreciated.

Answer 1

See maybe the article "Monotone matrices and monotone Markov processes" by Keilson and Kester (1977), and more precisely the discussion after Theorem 1.4.