Asymptotic behaviour of the non-extinction probability for a critical branching process

Critical age-dependent branching process with $k$ types $N_1,N_2,\dots,N_k$ of particles are considered. We suppose that particles' reproduction power depends on their age. Let $z_j^i(t)$ be the number of particles of type $N_j$ at time $t$ given that at time $t=0$ there was only one particle of type $N_i$. We derive an asymptotic formula for the probability $\mathbf P\{z_1^i(t)+z_2^i(t)+\dots+z_k^i(t)>0\}$ as $t\to\infty$. The result obtained is analogous to that of Goldstein [2].