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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.18919 |
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| _version_ | 1866915811094953984 |
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| author | Zhang, Zhenyuan |
| author_facet | Zhang, Zhenyuan |
| contents | Consider a discrete-time supercritical discounted branching random walk, in which increments at depth $k$ are independent and identically distributed with the same law as $m^{-kH}Y$, where $Y$ has a fixed law, $H>0$, and $m>1$ is the expected number of offspring at depth one. We provide a clean characterization of the boundedness of the discounted branching random walk: under mild conditions on the offspring distribution, the process is almost surely bounded if and only if $\mathbb{E}[|Y|^{1/H}]<\infty$. This extends results of Athreya (1985) and Aïdékon--Hu--Shi (2024), and provides a partial answer to Open Problem 31 of Aldous--Bandyopadhyay (2005). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18919 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Boundedness of discounted branching random walks via generic chaining Zhang, Zhenyuan Probability Consider a discrete-time supercritical discounted branching random walk, in which increments at depth $k$ are independent and identically distributed with the same law as $m^{-kH}Y$, where $Y$ has a fixed law, $H>0$, and $m>1$ is the expected number of offspring at depth one. We provide a clean characterization of the boundedness of the discounted branching random walk: under mild conditions on the offspring distribution, the process is almost surely bounded if and only if $\mathbb{E}[|Y|^{1/H}]<\infty$. This extends results of Athreya (1985) and Aïdékon--Hu--Shi (2024), and provides a partial answer to Open Problem 31 of Aldous--Bandyopadhyay (2005). |
| title | Boundedness of discounted branching random walks via generic chaining |
| topic | Probability |
| url | https://arxiv.org/abs/2602.18919 |