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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, "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0901.0264","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-01-02T18:07:35Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"d0f0b5c9d208821b0e78f5a7b15307ee78bae22a39ed58607c7146f3ed36f9ef","abstract_canon_sha256":"8194978aaf70fcc4cf5a5a2e70b560ba6522a71204bedc85492d6d80b8a49de4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:14.491791Z","signature_b64":"whWCRSpjnzi0V26BzQLGe/y21i/vbTZzrhIq0eRTpEYt7Uuvn04lFHIVbZTABBA0rL/MUqmhRdOrl0aDrBAyDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"416cc5739b64f1837d20ef94684e12a134612b975196dfd4449a384f9529806e","last_reissued_at":"2026-05-18T03:33:14.491245Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:14.491245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0901.0264","created_at":"2026-05-18T03:33:14.491344+00:00"},{"alias_kind":"arxiv_version","alias_value":"0901.0264v2","created_at":"2026-05-18T03:33:14.491344+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0264","created_at":"2026-05-18T03:33:14.491344+00:00"},{"alias_kind":"pith_short_12","alias_value":"IFWMK443MTYY","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"IFWMK443MTYYG7JA","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"IFWMK443","created_at":"2026-05-18T12:26:00.592388+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE","json":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE.json","graph_json":"https://pith.science/api/pith-number/IFWMK443MTYYG7JA56KGQTQSUE/graph.json","events_json":"https://pith.science/api/pith-number/IFWMK443MTYYG7JA56KGQTQSUE/events.json","paper":"https://pith.science/paper/IFWMK443"},"agent_actions":{"view_html":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE","download_json":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE.json","view_paper":"https://pith.science/paper/IFWMK443","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0901.0264&json=true","fetch_graph":"https://pith.science/api/pith-number/IFWMK443MTYYG7JA56KGQTQSUE/graph.json","fetch_events":"https://pith.science/api/pith-number/IFWMK443MTYYG7JA56KGQTQSUE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE/action/storage_attestation","attest_author":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE/action/author_attestation","sign_citation":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE/action/citation_signature","submit_replication":"https://pith.science/pith/IFWMK443MTYYG7JA56KGQTQSUE/action/replication_record"}},"created_at":"2026-05-18T03:33:14.491344+00:00","updated_at":"2026-05-18T03:33:14.491344+00:00"}