  A powerful tool for studying long-term convergence of a Markov process to its
stationary distribution is a Lyapunov function. In some sense, this is a
substitute for eigenfunctions. For a stochastically ordered Markov process on
the half-line, Lyapunov functions can be used to easily find explicit rates of
convergence. Our earlier research focused on exponential rate of convergence.
This note extends these results to slower rates, including power rates, thus
improving results of (Douc, Fort, Guillin, 2009).
