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

Critical age-dependent branching processes with $n$ types $T_1,\dots,T_n$ of particles are considered. Let $\mu_j^i(t)$ be the number of particles of type $T_j$ at time $t$ given that at time $t=0$ there was only one particle of type $T_i$. We derive an asymptotic formula for the probability $\mathbf P\{\mu_1^i(t)+\dots+ \mu_n^i(t)>0\}$ as $t\to\infty$.