{"paper":{"title":"The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Dan Cheng, Yimin Xiao","submitted_at":"2012-11-28T18:21:43Z","abstract_excerpt":"Let $X=\\{X(t),t\\in {\\mathbb{R}}^N\\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\\subset {\\mathbb{R}}^N$ and $u\\in {\\mathbb{R}}$, denote by $A_u=\\{t\\in T:X(t)\\geq u\\}$ the excursion set. Under $X(\\cdot)\\in C^2({\\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\\mathbb{E}}\\{\\varphi(A_u)\\}$, is derived. By applying the Rice method, it is shown that, as $u\\to\\infty$, the excursion probability ${\\mathbb{P}}\\{\\sup_{t\\in T}X(t)\\geq u\\}$ can be approximated by ${\\mathbb{E}}\\{\\varphi(A_u)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6693","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}