Properties of critical branching random walks on the line under non-extinction condition
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Valentin A. Topchii
Abstract
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.
Originally published in Diskretnaya Matematika (2023) 35, №1, 107–127 (in Russian).
Funding statement: The research was supported in accordance with the State assignment of the IM SB RAS, project FWNF-2022-0003.
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- On the number of particles from a marked set of cells for an analogue of a general allocation scheme
- Scaling of graphs with diameter constraint
- On the maximal tree in Galton–Watson forest with infinite variance of the offspring
- Short tests for contact circuits with similar-type weakly connected faults of contacts
- Large deviations of bisexual branching process in random environment
- Properties of critical branching random walks on the line under non-extinction condition
Artikel in diesem Heft
- Frontmatter
- On the number of particles from a marked set of cells for an analogue of a general allocation scheme
- Scaling of graphs with diameter constraint
- On the maximal tree in Galton–Watson forest with infinite variance of the offspring
- Short tests for contact circuits with similar-type weakly connected faults of contacts
- Large deviations of bisexual branching process in random environment
- Properties of critical branching random walks on the line under non-extinction condition
