A Conditional Functional Limit Theorem for Decomposable Branching Processes with Two Types of Particles

Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a functional limit theorem describing the distribution for the total number of particles of the second type appearing in the process in time $Nt$, $0\leq t<\infty$, given that the number of particles of the first type appearing in the process during its evolution is $N$.