Decomposable branching processes having a fixed extinction moment

The asymptotic behavior, as $n\rightarrow \infty$ of the probability of the event that a decomposable critical branching process $\mathbf{Z}(m)=(Z_{1}(m),...,Z_{N}(m)),$ $m=0,1,2,...,$ with $N$ types of particles dies at moment $n$ is investigated and conditional limit theorems are proved describing the distribution of the number of particles in the process $\mathbf{Z}(\cdot)$ at moment $m<n,$ given that the extinction moment of the process is $n$. These limit theorems may be considered as the statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees having a fixed hight.

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