Right-tail asymptotics for products of independent normal random variables
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915988570636288 |
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| author | Chvoinikov, Džiugas Šiaulys, Jonas |
| author_facet | Chvoinikov, Džiugas Šiaulys, Jonas |
| contents | Let $X_1,\dots,X_n$ be independent normal random variables with $X_i\sim N(μ_i,σ_i^2)$, and set $Z=\prod_{i=1}^n X_i$. We derive asymptotic approximations for the right tail probability $\mathbb{P}(Z>x)$ as $x\to\infty$. When at least one mean is nonzero, the asymptotic formula remains explicit and involves a finite multiplicative factor arising from admissible sign patterns (reflecting the different ways the product can be positive); it includes an explicit first relative correction term of order $x^{-1/n}$, with remaining relative error $O(x^{-2/n})$. The proof uses a boundary saddle-point/Laplace method: first a multidimensional Laplace approximation near the boundary saddle, then a one-dimensional endpoint Laplace approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08570 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Right-tail asymptotics for products of independent normal random variables Chvoinikov, Džiugas Šiaulys, Jonas Probability Let $X_1,\dots,X_n$ be independent normal random variables with $X_i\sim N(μ_i,σ_i^2)$, and set $Z=\prod_{i=1}^n X_i$. We derive asymptotic approximations for the right tail probability $\mathbb{P}(Z>x)$ as $x\to\infty$. When at least one mean is nonzero, the asymptotic formula remains explicit and involves a finite multiplicative factor arising from admissible sign patterns (reflecting the different ways the product can be positive); it includes an explicit first relative correction term of order $x^{-1/n}$, with remaining relative error $O(x^{-2/n})$. The proof uses a boundary saddle-point/Laplace method: first a multidimensional Laplace approximation near the boundary saddle, then a one-dimensional endpoint Laplace approximation. |
| title | Right-tail asymptotics for products of independent normal random variables |
| topic | Probability |
| url | https://arxiv.org/abs/2603.08570 |