On a class of branching processes

We study $\lim\limits_{n\to\infty}\mathbf P\{z_n=0\}$ (the probability of “degeneration”) where
1) $z_n=\sum\limits_{k=1}^{[z_{n-1}/a]}\xi_k+z_{n-1}-a[z_{n-1}/a]$, $n\ge1$
2) $a$ is a positive integer;
3) $\xi_n\ge0$ $(n\ge1)$ is a sequence of independent identically distributed integer-valued random variables.
If $a=1$, the sequence $\{z_n,n\ge0\}$ is an usual Galton–Watson branching process.