{"ID":2894338,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.11154","arxiv_id":"2507.11154","title":"Tube formula for spherically contoured random fields with subexponential marginals","abstract":"It is widely known that the tube method, or equivalently the Euler characteristic heuristic, provides a very accurate approximation for the tail probability that the supremum of a smooth Gaussian random field exceeds a threshold value $c$. The relative approximation error $Δ(c)$ is exponentially small as a function of $c$ when $c$ tends to infinity. On the other hand, little is known about non-Gaussian random fields. In this paper, we obtain the approximation error of the tube method applied to the canonical isotropic random fields on a unit sphere defined by $u\\mapsto\\langle u,ξ\\rangle$, $u\\in M\\subset\\mathbb{S}^{n-1}$, where $ξ$ is a spherically contoured random vector. These random fields have statistical applications in multiple testing and simultaneous regression inference when the unknown variance is estimated. The decay rate of the relative error $Δ(c)$ depends on the tail of the distribution of $\\|ξ\\|^2$ and the critical radius of the index set $M$. If this distribution is subexponential but not regularly varying, $Δ(c)\\to 0$ as $c\\to\\infty$. However, in the regularly varying case, $Δ(c)$ does not vanish and hence is not negligible. To address this limitation, we provide simple upper and lower bounds for $Δ(c)$ and for the tube formula itself. Numerical studies are conducted to assess the accuracy of the asymptotic approximation.","short_abstract":"It is widely known that the tube method, or equivalently the Euler characteristic heuristic, provides a very accurate approximation for the tail probability that the supremum of a smooth Gaussian random field exceeds a threshold value $c$. The relative approximation error $Δ(c)$ is exponentially small as a function of...","url_abs":"https://arxiv.org/abs/2507.11154","url_pdf":"https://arxiv.org/pdf/2507.11154v2","authors":"[\"Satoshi Kuriki\",\"Evgeny Spodarev\"]","published":"2025-07-15T10:04:16Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}