Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908893461872640 |
|---|---|
| author | Stonner, Jakob |
| author_facet | Stonner, Jakob |
| contents | We prove a version of Nagaev's theorem for the branching random walk with heavy-tailed associated random walk. For a branching random walk on $\mathbb{R}$ we consider the random measure $Z_n = \sum_{|u|=n} e^{-V_u} δ_{V_u}$ where $V_u$, $|u| = n$ denote the positions of the particles in the $n$-th generation. Under the assumption that $\mathbb{E}[Z_1(\cdot)]$ is a probability distribution with regularly varying tail, we prove that $Z_n((n\mathbb{E}[X] + t_n , \infty)) = W n\mathbb{P}(X > t_n )(1 + o(1))$ in $L^1$ as $n \to \infty$ where $W$ is a non-zero random variable, $t_n \uparrow \infty$ grows suitably fast, and $X$ has law $\mathbb{E}[Z_1(\cdot)]$. The result is explained probabilistically by a principle of one big jump for the branching random walk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16316 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump Stonner, Jakob Probability 60J80, 60F10 We prove a version of Nagaev's theorem for the branching random walk with heavy-tailed associated random walk. For a branching random walk on $\mathbb{R}$ we consider the random measure $Z_n = \sum_{|u|=n} e^{-V_u} δ_{V_u}$ where $V_u$, $|u| = n$ denote the positions of the particles in the $n$-th generation. Under the assumption that $\mathbb{E}[Z_1(\cdot)]$ is a probability distribution with regularly varying tail, we prove that $Z_n((n\mathbb{E}[X] + t_n , \infty)) = W n\mathbb{P}(X > t_n )(1 + o(1))$ in $L^1$ as $n \to \infty$ where $W$ is a non-zero random variable, $t_n \uparrow \infty$ grows suitably fast, and $X$ has law $\mathbb{E}[Z_1(\cdot)]$. The result is explained probabilistically by a principle of one big jump for the branching random walk. |
| title | Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump |
| topic | Probability 60J80, 60F10 |
| url | https://arxiv.org/abs/2603.16316 |