In this talk, I will present part of my thesis work carried out under the supervision of Bastien Mallein and Laurent Tournier. We are interested in a Markov process in continuous time and discrete state space reaching an absorbing state in finite time. The study of its behavior before absorption is strongly linked to the study of its quasi-stationary distributions for which a known necessary condition of existence is the so-called ‘exponential killing’ of the process. After introducing the theory of quasi-stationary measures in discrete state space, I will present an original framework in which the exponential killing condition is sufficient for the existence of the Yaglom limit of the process. In particular, we assume that the process can only reach its absorbing state from a single other state, called the ‘exit’ state, and then propose to decompose its trajectories into excursions out of this exit state. This trajectory approach enables us to link the study of the Yaglom limit of the process to the asymptotic behavior of the inverse of its local time in the exit state. Finally, we obtain a representation of the Yaglom limit from the excursion measure of the process.