The Structure of the Decomposable Reduced Branching Processes. I. Finite-Dimensional Distributions

A decomposable Galton--Watson branching process with $N$ types of particles labeled $1,2,\ldots,N$ is considered in which a type $i$ parent may produce individuals of types $j\geq i$ only. This model may be viewed as a stochastic model of the development of a population whose individuals may be located at one of the $N$ islands, the location of a particle being considered as its type. The newborn particles of island $i\le N-1$ either stay at the same island or migrate, just after their birth to the islands $i+1,i+2,\ldots,N$. Particles of island $N$ do not migrate. Let $Z_{i}(m,n)$ be the number of type $i$ particles existing in the process at moment $mGalton--Watson process is strongly critical we investigate properties of the finite-dimensional distributions of the process $ {\bf Z}(m,n)=(Z_{1}(m,n),\ldots,Z_{N}(m,n)) $ dependent on the growth rate of the parameter $m=m(n)$ as $n\to\infty.$