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Consider the approximation of the self-intersection local time of $B$, defined as \\begin{align*} I_{T}^{\\varepsilon}\n  &=\\int_{0}^{T}\\int_{0}^{t}p_{\\varepsilon}(B_{t}-B_{s})dsdt, \\end{align*} where $p_\\varepsilon(x)$ is the heat kernel. We prove that the process $\\{I_{T}^{\\varepsilon}-\\mathbb{E}\\left[I_{T}^{\\varepsilon}\\right]\\}_{T\\geq0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\\frac{3}{2d}<H\\leq\\frac{3}"},"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":"1701.05289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-01-19T03:56:27Z","cross_cats_sorted":[],"title_canon_sha256":"6d757d99e33cf93e56b67fce43ff1f7884d7070f938ea2f02bc9db162dd60456","abstract_canon_sha256":"92faa52b4959e81319b211531c79191b4294c2666038ff2965e223883e8a3063"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:30.336078Z","signature_b64":"A0bMQlQQf88x7Ulxe/O/R2uCRfnQO82A0+yHXxSTJkTKoxdOxATz58Q7pUgfk+JWVlymSHE60UXoivQNMzsjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b939ea9587381f820a881250211631a59e8776efe259c78e3e526e0f8d02b5f9","last_reissued_at":"2026-05-18T00:52:30.335523Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:30.335523Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Functional limit theorem for the self-intersection local time of the fractional Brownian motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arturo Jaramillo, David Nualart","submitted_at":"2017-01-19T03:56:27Z","abstract_excerpt":"Let $\\{B_{t}\\}_{t\\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $0<H<1$, where $d\\geq2$. Consider the approximation of the self-intersection local time of $B$, defined as \\begin{align*} I_{T}^{\\varepsilon}\n  &=\\int_{0}^{T}\\int_{0}^{t}p_{\\varepsilon}(B_{t}-B_{s})dsdt, \\end{align*} where $p_\\varepsilon(x)$ is the heat kernel. We prove that the process $\\{I_{T}^{\\varepsilon}-\\mathbb{E}\\left[I_{T}^{\\varepsilon}\\right]\\}_{T\\geq0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\\frac{3}{2d}<H\\leq\\frac{3}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05289","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":"1701.05289","created_at":"2026-05-18T00:52:30.335617+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.05289v1","created_at":"2026-05-18T00:52:30.335617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.05289","created_at":"2026-05-18T00:52:30.335617+00:00"},{"alias_kind":"pith_short_12","alias_value":"XE46VFMHHAPY","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"XE46VFMHHAPYECUI","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"XE46VFMH","created_at":"2026-05-18T12:31:53.515858+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/XE46VFMHHAPYECUICJICCFRRUW","json":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW.json","graph_json":"https://pith.science/api/pith-number/XE46VFMHHAPYECUICJICCFRRUW/graph.json","events_json":"https://pith.science/api/pith-number/XE46VFMHHAPYECUICJICCFRRUW/events.json","paper":"https://pith.science/paper/XE46VFMH"},"agent_actions":{"view_html":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW","download_json":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW.json","view_paper":"https://pith.science/paper/XE46VFMH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.05289&json=true","fetch_graph":"https://pith.science/api/pith-number/XE46VFMHHAPYECUICJICCFRRUW/graph.json","fetch_events":"https://pith.science/api/pith-number/XE46VFMHHAPYECUICJICCFRRUW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW/action/storage_attestation","attest_author":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW/action/author_attestation","sign_citation":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW/action/citation_signature","submit_replication":"https://pith.science/pith/XE46VFMHHAPYECUICJICCFRRUW/action/replication_record"}},"created_at":"2026-05-18T00:52:30.335617+00:00","updated_at":"2026-05-18T00:52:30.335617+00:00"}