  Consider a finite absorbing Markov generator, irreducible on the
non-absorbing states. Perron-Frobenius theory ensures the existence of a
corresponding positive eigenvector $\varphi$. The goal of the paper is to give
bounds on the amplitude $\max \varphi/\min\varphi$. Two approaches are
proposed: one using a path method and the other one, restricted to the
reversible situation, based on spectral estimates. The latter approach is
extended to denumerable birth and death processes absorbing at 0 for which
infinity is an entrance boundary. The interest of estimating the ratio is the
reduction of the quantitative study of convergence to quasi-stationarity to the
convergence to equilibrium of related ergodic processes, as seen in [7].
