Minimum and extremal process for a branching random walk outside the boundary case
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912819480363008 |
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| author | Chen, Xinxin Hou, Haojie |
| author_facet | Chen, Xinxin Hou, Haojie |
| contents | This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function explodes at $1$ and the random walk associated to the spine possesses a stretched exponential tail with exponent $b\in(0,\frac12)$. Under suitable conditions, we confirm the conjecture of Barral, Hu and Madaule [Bernoulli 24(2) 2018 801-841], and obtain the weak convergence for the minimum and the extremal process. We also establish an a.s. infimum result over all infinity rays of this system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07129 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimum and extremal process for a branching random walk outside the boundary case Chen, Xinxin Hou, Haojie Probability This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function explodes at $1$ and the random walk associated to the spine possesses a stretched exponential tail with exponent $b\in(0,\frac12)$. Under suitable conditions, we confirm the conjecture of Barral, Hu and Madaule [Bernoulli 24(2) 2018 801-841], and obtain the weak convergence for the minimum and the extremal process. We also establish an a.s. infimum result over all infinity rays of this system. |
| title | Minimum and extremal process for a branching random walk outside the boundary case |
| topic | Probability |
| url | https://arxiv.org/abs/2601.07129 |