Asymptotic formulas for products of Poisson distributions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918426061045760 |
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| author | Chvoinikov, Džiugas Šiaulys, Jonas |
| author_facet | Chvoinikov, Džiugas Šiaulys, Jonas |
| contents | In this paper, we study the asymptotic behaviour of the product tail probability $
\mathbb{P}(ξ_1\cdotsξ_N \geqslant n), $
where $\{ξ_1,\ldots,ξ_N\}$ is a finite collection of independent Poisson random variables with positive parameters $λ_1,\ldots,λ_N$. We derive a refined Laplace-type asymptotic formula for the tail probability, based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert function, and a careful evaluation of the constrained Gaussian prefactor. This yields an explicit approximation with an $O(\log n)$ remainder term in the exponent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08598 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic formulas for products of Poisson distributions Chvoinikov, Džiugas Šiaulys, Jonas Probability In this paper, we study the asymptotic behaviour of the product tail probability $ \mathbb{P}(ξ_1\cdotsξ_N \geqslant n), $ where $\{ξ_1,\ldots,ξ_N\}$ is a finite collection of independent Poisson random variables with positive parameters $λ_1,\ldots,λ_N$. We derive a refined Laplace-type asymptotic formula for the tail probability, based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert function, and a careful evaluation of the constrained Gaussian prefactor. This yields an explicit approximation with an $O(\log n)$ remainder term in the exponent. |
| title | Asymptotic formulas for products of Poisson distributions |
| topic | Probability |
| url | https://arxiv.org/abs/2603.08598 |