Decomposable branching processes with a fixed extinction moment

V. A. Vatutin ; E. E. D'yakonova

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 114-135

Résumé

The asymptotic behavior as $n\to \infty $ of the probability of the event that a decomposable critical branching process $\mathbf Z(m)= (Z_1(m),\dots ,Z_N(m))$, $m=0,1,2,\dots $, with $N$ types of particles dies at moment $n$ is investigated, and conditional limit theorems are proved that describe the distribution of the number of particles in the process $\mathbf Z(\cdot )$ at moment $m$ given that the extinction moment of the process is $n$. These limit theorems can be considered as statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees of fixed height.