On local large deviations for decoupled random walks

[2508.05178] On local large deviations for decoupled random walks Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2508.05178 (math) [Submitted on 7 Aug 2025 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: On local large deviations for decoupled random walks Authors: Dariusz Buraczewski , Alexander Iksanov , Alexander Marynych View PDF HTML (experimental) Abstract: 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 $S_n = \xi_1 + \ldots + \xi_n$, where $(\xi_k)_{k \geq 1}$ are independent copies of a nonnegative random variable $\xi$. 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 $\xi$, 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. Subjects: Probability (math.PR) MSC classes: Primary: 60F10, 60F05, Secondary: 60G55 Cite as: arXiv:2508.05178 [math.PR] (or arXiv:2508.05178v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.2508.05178 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Alexander Marynych [ view email ] [v1] Thu, 7 Aug 2025 09:15:08 UTC (36 KB) [v2] Wed, 26 Aug 2026 19:43:39 UTC (38 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2025-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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