Branching random walk conditioned on large martingale limit
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908330467786752 |
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| author | Chen, Xinxin de Raphélis, Loïc Ma, Heng |
| author_facet | Chen, Xinxin de Raphélis, Loïc Ma, Heng |
| contents | We consider a branching random walk in the non-boundary case where the additive martingale $W_n$ converges a.s. and in mean to some non-degenerate limit $W_\infty$. We first establish the joint tail distribution of $W_\infty$ and the global minimum of this branching random walk. Next, conditioned on the event that the minimum is atypically small or conditioned on very large $W_\infty$, we study the branching random walk viewed from the minimum and obtain the convergence in law in the vague sense. As a byproduct, we also get the right tail of the limit of derivative martingale. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05538 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Branching random walk conditioned on large martingale limit Chen, Xinxin de Raphélis, Loïc Ma, Heng Probability We consider a branching random walk in the non-boundary case where the additive martingale $W_n$ converges a.s. and in mean to some non-degenerate limit $W_\infty$. We first establish the joint tail distribution of $W_\infty$ and the global minimum of this branching random walk. Next, conditioned on the event that the minimum is atypically small or conditioned on very large $W_\infty$, we study the branching random walk viewed from the minimum and obtain the convergence in law in the vague sense. As a byproduct, we also get the right tail of the limit of derivative martingale. |
| title | Branching random walk conditioned on large martingale limit |
| topic | Probability |
| url | https://arxiv.org/abs/2408.05538 |