Boundedness of discounted tree sums
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910990691467264 |
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| author | Aïdékon, Elie Hu, Yueyun Shi, Zhan |
| author_facet | Aïdékon, Elie Hu, Yueyun Shi, Zhan |
| contents | Let $(V(u),\, u\in \mathcal{T})$ be a (supercritical) branching random walk and $(η_u,\,u\in \mathcal{T})$ be marks on the vertices of the tree, distributed in an i.i.d.\ fashion. Following Aldous and Bandyopadhyay \cite{AB05}, for each infinite ray $ξ$ of the tree, we associate the {\it discounted tree sum} $D(ξ)$ which is the sum of the $e^{-V(u)}η_u$ taken along the ray. The paper deals with the finiteness of $\sup_ξD(ξ)$. To this end, we study the extreme behaviour of the local time processes of the paths $(V(u),\,u\in ξ)$. It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay \cite{AB05}. We also present several open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01048 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundedness of discounted tree sums Aïdékon, Elie Hu, Yueyun Shi, Zhan Probability Let $(V(u),\, u\in \mathcal{T})$ be a (supercritical) branching random walk and $(η_u,\,u\in \mathcal{T})$ be marks on the vertices of the tree, distributed in an i.i.d.\ fashion. Following Aldous and Bandyopadhyay \cite{AB05}, for each infinite ray $ξ$ of the tree, we associate the {\it discounted tree sum} $D(ξ)$ which is the sum of the $e^{-V(u)}η_u$ taken along the ray. The paper deals with the finiteness of $\sup_ξD(ξ)$. To this end, we study the extreme behaviour of the local time processes of the paths $(V(u),\,u\in ξ)$. It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay \cite{AB05}. We also present several open questions. |
| title | Boundedness of discounted tree sums |
| topic | Probability |
| url | https://arxiv.org/abs/2409.01048 |