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 $p_k$, $k\ge0$. In this paper we consider {\it defective} Galton-Watson processes having defective reproduction laws, so that $\sum_{k\ge0}p_k=1-\eps$ for some $\eps\in(0,1)$. In this setting, each particle may send the process to a graveyard state $\Delta$ with probability $\eps$. Such a Markov chain, having an enhanced state space $\{0,1,\ldots\}\cup\{\Delta\}$, gets eventually absorbed either at $0$ or at $\Delta$. Assuming that the process has avoided absorption until the observation time $t$, we are interested in its trajectories as $t\to\infty$ and $\eps\to0$.