Upper deviation probabilities for level sets of a supercritical branching random walk
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910319523135488 |
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| author | Zhang, Shuxiong Luo, Lianghui |
| author_facet | Zhang, Shuxiong Luo, Lianghui |
| contents | Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of $\{Z_n\}_{n\geq 0}$ and $I(x)$ be the large deviation rate function of the underlying random walk of $\{Z_n\}_{n\geq 0}$. It is known from [6] that under some mild conditions, for $x\in(0,x^*)$, $n^{-1}\log Z_n([nx,\infty))$ converges almost surely to $\log m- I(x)$ on the event of nonextinction as $n\to\infty$, where $x^*$ is the speed of maximal position of the branching random walk. In this work, we investigate its upper deviation probabilities, in other words, the convergence rates of \[\mathbb{P}(Z_n([xn,\infty))\geq e^{an})\] as $n\to\infty$, where $x>0$ and $a>(\log m- I(x))^+$. This paper is a counterpart work of the lower deviation probabilities [28] and also completes those results in [1] for the branching Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03872 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper deviation probabilities for level sets of a supercritical branching random walk Zhang, Shuxiong Luo, Lianghui Probability Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of $\{Z_n\}_{n\geq 0}$ and $I(x)$ be the large deviation rate function of the underlying random walk of $\{Z_n\}_{n\geq 0}$. It is known from [6] that under some mild conditions, for $x\in(0,x^*)$, $n^{-1}\log Z_n([nx,\infty))$ converges almost surely to $\log m- I(x)$ on the event of nonextinction as $n\to\infty$, where $x^*$ is the speed of maximal position of the branching random walk. In this work, we investigate its upper deviation probabilities, in other words, the convergence rates of \[\mathbb{P}(Z_n([xn,\infty))\geq e^{an})\] as $n\to\infty$, where $x>0$ and $a>(\log m- I(x))^+$. This paper is a counterpart work of the lower deviation probabilities [28] and also completes those results in [1] for the branching Brownian motion. |
| title | Upper deviation probabilities for level sets of a supercritical branching random walk |
| topic | Probability |
| url | https://arxiv.org/abs/2402.03872 |