Extinction of decomposable branching processes

The asymptotic behavior, as $n\rightarrow \infty $, of the conditional distribution of the number of particles in a decomposable critical branching process $\mathbf{Z}(m)=(Z_{1}(m),\ldots,Z_{N}(m))$ with $N$ types of particles at moment $m=n-k,\, k=o(n)$, is investigated given that the extinction moment of the process equals to $n$.