On local large deviations for decoupled random walks
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915433516367872 |
|---|---|
| author | Buraczewski, Dariusz Iksanov, Alexander Marynych, Alexander |
| author_facet | Buraczewski, Dariusz Iksanov, Alexander Marynych, Alexander |
| contents | A decoupled standard random walk is a sequence of independent random variables $(\hat{S}_n)_{n \geq 1}$ such that, for each $n \geq 1$, the distribution of $\hat{S}_n$ is the same as that of $S_n = ξ_1 + \ldots + ξ_n$, where $(ξ_k)_{k \geq 1}$ are independent copies of a nonnegative random variable $ξ$. We consider the counting process $(\hat{N}(t))_{t\geq 0}$ defined as the number of terms $\hat{S}_n$ in the sequence $(\hat{S}_n)_{n \geq 1}$ that lie within the interval $[0, t]$. Under various assumptions on the tail distribution of $ξ$, we derive logarithmic asymptotics for the local large deviation probabilities $\mathbb{P}\{\hat{N}(t) = \lfloor b \, \mathbb{E}[\hat{N}(t)] \rfloor\}$ as $t \to \infty$ for a fixed constant $b > 0$. These results are then applied to obtain a logarithmic local large deviations asymptotic for the counting process associated with the infinite Ginibre ensemble and, more generally, for determinantal point processes with the Mittag-Leffler kernel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On local large deviations for decoupled random walks Buraczewski, Dariusz Iksanov, Alexander Marynych, Alexander Probability Primary: 60F10, 60F05, Secondary: 60G55 A decoupled standard random walk is a sequence of independent random variables $(\hat{S}_n)_{n \geq 1}$ such that, for each $n \geq 1$, the distribution of $\hat{S}_n$ is the same as that of $S_n = ξ_1 + \ldots + ξ_n$, where $(ξ_k)_{k \geq 1}$ are independent copies of a nonnegative random variable $ξ$. We consider the counting process $(\hat{N}(t))_{t\geq 0}$ defined as the number of terms $\hat{S}_n$ in the sequence $(\hat{S}_n)_{n \geq 1}$ that lie within the interval $[0, t]$. Under various assumptions on the tail distribution of $ξ$, we derive logarithmic asymptotics for the local large deviation probabilities $\mathbb{P}\{\hat{N}(t) = \lfloor b \, \mathbb{E}[\hat{N}(t)] \rfloor\}$ as $t \to \infty$ for a fixed constant $b > 0$. These results are then applied to obtain a logarithmic local large deviations asymptotic for the counting process associated with the infinite Ginibre ensemble and, more generally, for determinantal point processes with the Mittag-Leffler kernel. |
| title | On local large deviations for decoupled random walks |
| topic | Probability Primary: 60F10, 60F05, Secondary: 60G55 |
| url | https://arxiv.org/abs/2508.05178 |