Defective Galton-Watson processes

The Galton-Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p(k), k 0. In this paper, we consider defective Galton-Watson processes having defective reproduction laws, so that Sigma(k 0)p(k) = 1 - E for some E (0, 1). In this setting, each particle may send the process to a graveyard state with probability E. Such a Markov chain, having an enhanced state space {0, 1, ...}{}, gets eventually absorbed either at 0 or at . Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t and E 0