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Here $\\xi_{\\vec{h}}(x), x\\in(-b,b)^d, d\\geq 1$ is a kernel-type gaussian random field and $\\|\\cdot\\|_p$ stands for $L_p$-norm on $(-b,b)^d$. The set $\\mathrm{H}$ consists of $d$-variate vector-functions defined on $(-b,b)^d$ and taking values in some countable net in $R^d_+$. We seek a non-random family $\\left\\{\\Psi_\\alpha\\big(\\vec{h}\\big),\\;\\;\\vec{h}\\in\\mathrm{H}\\right\\}$ such that $ E\\big\\{\\sup_{\\vec{h}\\in\\mathrm{H}}\\"},"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":"1311.4996","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-20T10:07:02Z","cross_cats_sorted":[],"title_canon_sha256":"124129c9ab5d22590e0e315aa9113848daf861dfb00809d0ab137d52ab0f9c3b","abstract_canon_sha256":"6a1cc989d4ef78f059fa0872fc2f9d9a75a9c53f06db34dca17d55b79e23d708"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:42.245675Z","signature_b64":"pZrqfCQjSy+grzSj1d5jrUZHrlKRsXK6eO1JaV4hUf5rn3d98yKGBC8KtWpBmGPKYyUsIUIVwR0uSQZyqs6OBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98adbfea4dbbd14feb88cdd2c490286a6551e256453fd663207e793cdcefb56b","last_reissued_at":"2026-05-18T03:06:42.245194Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:42.245194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper functions for $L_p$-norm of gaussian random fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"O. Lepski","submitted_at":"2013-11-20T10:07:02Z","abstract_excerpt":"In this paper we are interested in finding upper functions for a collection of random variables $\\big\\{\\big\\|\\xi_{\\vec{h}}\\big\\|_p, \\vec{h}\\in\\mathrm{H}\\big\\}, 1\\leq p<\\infty$. Here $\\xi_{\\vec{h}}(x), x\\in(-b,b)^d, d\\geq 1$ is a kernel-type gaussian random field and $\\|\\cdot\\|_p$ stands for $L_p$-norm on $(-b,b)^d$. The set $\\mathrm{H}$ consists of $d$-variate vector-functions defined on $(-b,b)^d$ and taking values in some countable net in $R^d_+$. We seek a non-random family $\\left\\{\\Psi_\\alpha\\big(\\vec{h}\\big),\\;\\;\\vec{h}\\in\\mathrm{H}\\right\\}$ such that $ E\\big\\{\\sup_{\\vec{h}\\in\\mathrm{H}}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4996","kind":"arxiv","version":1},"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":"1311.4996","created_at":"2026-05-18T03:06:42.245278+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.4996v1","created_at":"2026-05-18T03:06:42.245278+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4996","created_at":"2026-05-18T03:06:42.245278+00:00"},{"alias_kind":"pith_short_12","alias_value":"TCW372SNXPIU","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_16","alias_value":"TCW372SNXPIU724I","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_8","alias_value":"TCW372SN","created_at":"2026-05-18T12:27:59.945178+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/TCW372SNXPIU724IZXJMJEBINJ","json":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ.json","graph_json":"https://pith.science/api/pith-number/TCW372SNXPIU724IZXJMJEBINJ/graph.json","events_json":"https://pith.science/api/pith-number/TCW372SNXPIU724IZXJMJEBINJ/events.json","paper":"https://pith.science/paper/TCW372SN"},"agent_actions":{"view_html":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ","download_json":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ.json","view_paper":"https://pith.science/paper/TCW372SN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.4996&json=true","fetch_graph":"https://pith.science/api/pith-number/TCW372SNXPIU724IZXJMJEBINJ/graph.json","fetch_events":"https://pith.science/api/pith-number/TCW372SNXPIU724IZXJMJEBINJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ/action/storage_attestation","attest_author":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ/action/author_attestation","sign_citation":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ/action/citation_signature","submit_replication":"https://pith.science/pith/TCW372SNXPIU724IZXJMJEBINJ/action/replication_record"}},"created_at":"2026-05-18T03:06:42.245278+00:00","updated_at":"2026-05-18T03:06:42.245278+00:00"}