Moment asymptotics for branching processes close to critical ones

We consider an age-dependent branching process with $n$ different types of particles. Let $\mu_j^i(t)$ be the number of particles of type $j$ at time $t$ if the process started from a single particle of type $i$. Let $A$ be the matrix of the process, and let $R$ be the Perron eigenvalue of $A$. We obtain an asymptotics of $A_j^i(t)=\mathbf M\mu_j^i(t)$ and $B_{jk}^i(t)=\mathbf M\mu_j^i(t)(\mu_k^i(t)-\delta_j^k)$ as $t\to\infty$, $R\to 1$ in a compact matrix class $\mathfrak A=\{A\}$.