{"paper":{"title":"Representation of small ball probabilities in Hilbert space and lower bound in regression for functional data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Andr\\'e Mas (I3M)","submitted_at":"2009-01-02T18:07:35Z","abstract_excerpt":"Let $S=\\sum_{i=1}^{+\\infty}\\lambda_{i}Z_{i}$ where the $Z_{i}$'s are i.d.d. positive with $\\mathbb{E}\\| Z\\| ^{3}<+\\infty$ and $(\\lambda_{i})_{i\\in\\mathbb{N}}$ a positive nonincreasing sequence such that $\\sum\\lambda_{i}<+\\infty$. We study the small ball probability $\\mathbb{P}(S<\\epsilon) $ when $\\epsilon\\downarrow0$. We start from a result by Lifshits (1997) who computed this probability by means of the Laplace transform of $S$. We prove that $\\mathbb{P}(S<\\cdot) $ belongs to a class of functions introduced by de Haan, well-known in extreme value theory, the class of Gamma-varying functions, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0264","kind":"arxiv","version":2},"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"}