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