Geodesic
Properties of critical branching random walks on the line under non-extinction condition
Diskretnaya Matematika, Tome 35 (2023) no. 1, pp. 107-127
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We study a critical branching random walk on a real line with discrete time controlled by a point process. Sizes of sequential generations form Galton–Watson critical process with one type of particles. The particle coordinates are interpreted as the weights of the vertices on the genealogical tree of the random walk. A reduced tree is obtained after removing branches of the genealogical tree that do not reach the $n$-th level. The asymptotic behavior of the first two moments of the number of vertices and of the sum of vertex weights over all levels of reduced tree under condition of nonextinction is described. Several limit theorems for the weights of particles in a branching random walk up to $n$-th generation under condition of nonextinction are proved.
Keywords: branching random walk on a real line, Galton–Watson branching process, point process on a line, critical branching process, limit theorems, reduced genealogical trees. 
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V. A. Topchii. Properties of critical branching random walks on the line under non-extinction condition. Diskretnaya Matematika, Tome 35 (2023) no. 1, pp. 107-127. http://geodesic.mathdoc.fr/item/DM_2023_35_1_a7/

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