Geodesic
Limit Distributions for the Number of Particles in Branching Random Walks
Matematičeskie zametki, Tome 90 (2011) no. 6, pp. 845-859

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We study branching random walks with continuous time. Particles performing a random walk on $\mathbb{Z}^{2}$, are allowed to be born and die only at the origin. It is assumed that the offspring reproduction law at the branching source is critical and the random walk outside the source is homogeneous and symmetric. Given particles at the origin, we prove a conditional limit theorem for the joint distribution of suitably normalized numbers of particles at the source and outside it as time unboundedly increases. As a consequence, we establish the asymptotic independence of such random variables.
Keywords: branching random walk, branching source, offspring reproduction law, Bellman–Harris branching process, probability generating function
Mots-clés : transition rate matrix. 
@article{MZM_2011_90_6_a3,
     author = {E. Vl. Bulinskaya},
     title = {Limit {Distributions} for the {Number} of {Particles} in {Branching} {Random} {Walks}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {845--859},
     publisher = {mathdoc},
     volume = {90},
     number = {6},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_6_a3/}
}
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E. Vl. Bulinskaya. Limit Distributions for the Number of Particles in Branching Random Walks. Matematičeskie zametki, Tome 90 (2011) no. 6, pp. 845-859. http://geodesic.mathdoc.fr/item/MZM_2011_90_6_a3/