  In this paper we study Monte Carlo estimators based on the likelihood ratio
approach for steady-state sensitivity.
  We first extend the result of Glynn and Olvera-Cravioto [doi:doi:
10.1287/stsy.2018.002] to the setting of continuous time Markov chains with a
countable state space which include models such as stochastic reaction kinetics
and kinetic Monte Carlo lattice system.
  We show that the variance of the centered likelihood ratio estimators does
not grow in time. This result suggests that the centered likelihood ratio
estimators should be favored for sensitivity analysis when the mixing time of
the underlying continuous time Markov chain is large, which is typically the
case when systems exhibit multi-scale behavior. We demonstrate a practical
implication of this analysis on a numerical benchmark of two examples for the
biochemical reaction networks.
