Algebraic convergence in the $L^2$-sense is studied for general time-continuous, reversible Markov chains with countable state space, and especially for birth--death chains. Some criteria for the convergence are presented. The results are effective since the convergence region can be completely covered, as illustrated by two examples.
"Algebraic convergence of Markov chains." Ann. Appl. Probab. 13 (2) 604 - 627, May 2003. https://doi.org/10.1214/aoap/1050689596