Geodesic
Atypical population size in a~two-type decomposable branching process
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 649-671

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We consider a Galton–Watson branching process with particles of two types in which particles of type one produce both particles of types one and two, and particles of type two generate offsprings of only type two. It is known that if both types are critical, then, for a process that is initiated at time $0$ by a single type-one particle, the number of particles of type two at time $n$ (provided that the process is not degenerate by this time) is proportional to $n$. We find the asymptotics of the probability that the number of type-two particles at time $n$ is of the order $o(n) $ (provided that the process is not degenerate by this time).
Keywords: reduced branching process, population size, local limit theorem. 
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     author = {V. A. Vatutin and E. E. D'yakonova},
     title = {Atypical population size in a~two-type decomposable branching process},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {649--671},
     publisher = {mathdoc},
     volume = {67},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a1/}
}
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V. A. Vatutin; E. E. D'yakonova. Atypical population size in a~two-type decomposable branching process. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 4, pp. 649-671. http://geodesic.mathdoc.fr/item/TVP_2022_67_4_a1/