{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:DNTLMF54N3OHXFOLC6ZNNN4RKD","short_pith_number":"pith:DNTLMF54","canonical_record":{"source":{"id":"2604.19353","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-04-21T11:34:24Z","cross_cats_sorted":["stat.ME","stat.TH"],"title_canon_sha256":"7da79c1ae9c1e0d0e2b40c64b81e044822be740dac21c4e73b8500e50af00f26","abstract_canon_sha256":"a99900fa7e89bce8f474d939c7f9e40e245f3fb5f5fc8122463cb7b09dde4560"},"schema_version":"1.0"},"canonical_sha256":"1b66b617bc6edc7b95cb17b2d6b79150f267cf243aa1ab41503caf2eb6ee0aeb","source":{"kind":"arxiv","id":"2604.19353","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.19353","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"arxiv_version","alias_value":"2604.19353v2","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.19353","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"pith_short_12","alias_value":"DNTLMF54N3OH","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"pith_short_16","alias_value":"DNTLMF54N3OHXFOL","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"pith_short_8","alias_value":"DNTLMF54","created_at":"2026-05-25T02:02:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:DNTLMF54N3OHXFOLC6ZNNN4RKD","target":"record","payload":{"canonical_record":{"source":{"id":"2604.19353","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-04-21T11:34:24Z","cross_cats_sorted":["stat.ME","stat.TH"],"title_canon_sha256":"7da79c1ae9c1e0d0e2b40c64b81e044822be740dac21c4e73b8500e50af00f26","abstract_canon_sha256":"a99900fa7e89bce8f474d939c7f9e40e245f3fb5f5fc8122463cb7b09dde4560"},"schema_version":"1.0"},"canonical_sha256":"1b66b617bc6edc7b95cb17b2d6b79150f267cf243aa1ab41503caf2eb6ee0aeb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:02:15.579225Z","signature_b64":"gm0wiIeDyHNIwmwp6sIWYNpYhTOryt0En9O/Yo2mR+gHjwISONcJpCN2Y3ZZjQ8pav2L/LhsyWUU+8qjS84tBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b66b617bc6edc7b95cb17b2d6b79150f267cf243aa1ab41503caf2eb6ee0aeb","last_reissued_at":"2026-05-25T02:02:15.578541Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:02:15.578541Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.19353","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:02:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wd4zJw5OUbzqFsCqtoyWZkTWHJiHSL3aW2GB16lE3jMEqrd3qHHgsO8QUV6i9NjO4RkD1QMyk6G9zngGZC1WCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:39:19.143808Z"},"content_sha256":"30e1181c1da0be094d15da8d49550b39168b1c28b200025598c34b997f99a771","schema_version":"1.0","event_id":"sha256:30e1181c1da0be094d15da8d49550b39168b1c28b200025598c34b997f99a771"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:DNTLMF54N3OHXFOLC6ZNNN4RKD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic e-processes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality.","cross_cats":["stat.ME","stat.TH"],"primary_cat":"math.ST","authors_text":"Mattes Mollenhauer, Pierre-Fran\\c{c}ois Massiani, Sebastian Schulze","submitted_at":"2026-04-21T11:34:24Z","abstract_excerpt":"We investigate the concept of an asymptotic e-process, which is a doubly-indexed stochastic process $(E_{m,n})_{m,n\\in\\mathbb{N}}$ that possesses, asymptotically for an approximation index $m\\to\\infty$, the properties of an e-process along a monitoring time index $n$. This constitutes the first in-depth study of this recently introduced concept, which is relevant in asymptotic sequential anytime-valid inference. Our theory is motivated by practical applications in sequential hypothesis testing, in which e-variables and e-processes can only be constructed approximately from observations due to "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive an asymptotic version of Ville's inequality, which bounds excursion probabilities of (E_{m,n})_{m,n∈ℕ} over some threshold uniformly over n up to a time horizon r_m that is determined by the quality of process approximation over m.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The approximation quality of the doubly indexed process to an e-process as m→∞ is sufficient to determine a growing time horizon r_m over which the uniform bound holds, as stated in the abstract's description of the limiting behavior.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Asymptotic e-processes approximate e-processes for large m, enabling an asymptotic Ville's inequality that bounds uniform excursion probabilities up to a time horizon r_m determined by approximation quality.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a0c3954425e721992af4b5cb69ebf01e28da0f94faf70212fee8759adad34d9b"},"source":{"id":"2604.19353","kind":"arxiv","version":2},"verdict":{"id":"53ad64bb-6489-425a-a83f-1c07372d799f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T01:34:44.177933Z","strongest_claim":"We derive an asymptotic version of Ville's inequality, which bounds excursion probabilities of (E_{m,n})_{m,n∈ℕ} over some threshold uniformly over n up to a time horizon r_m that is determined by the quality of process approximation over m.","one_line_summary":"Asymptotic e-processes approximate e-processes for large m, enabling an asymptotic Ville's inequality that bounds uniform excursion probabilities up to a time horizon r_m determined by approximation quality.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The approximation quality of the doubly indexed process to an e-process as m→∞ is sufficient to determine a growing time horizon r_m over which the uniform bound holds, as stated in the abstract's description of the limiting behavior.","pith_extraction_headline":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.19353/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T03:04:23.787679Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"d29b5909a2e28753a251a729ac5210149a380e4041e23f26abad9d113919e273"},"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"},"verdict_id":"53ad64bb-6489-425a-a83f-1c07372d799f"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:02:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X6dmepPeujKHmy217Tpec/OAljZ0/7Xzy/FqNW7pFGygg3rynNRx4ZFuZzB7u+02NXGs8yAUEW6ldnxEkeoADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:39:19.144392Z"},"content_sha256":"5f7e920799daa177ed2d44aafe1f9303c30d7950b8fa06edfa8c9cbb78f692ae","schema_version":"1.0","event_id":"sha256:5f7e920799daa177ed2d44aafe1f9303c30d7950b8fa06edfa8c9cbb78f692ae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DNTLMF54N3OHXFOLC6ZNNN4RKD/bundle.json","state_url":"https://pith.science/pith/DNTLMF54N3OHXFOLC6ZNNN4RKD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DNTLMF54N3OHXFOLC6ZNNN4RKD/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T22:39:19Z","links":{"resolver":"https://pith.science/pith/DNTLMF54N3OHXFOLC6ZNNN4RKD","bundle":"https://pith.science/pith/DNTLMF54N3OHXFOLC6ZNNN4RKD/bundle.json","state":"https://pith.science/pith/DNTLMF54N3OHXFOLC6ZNNN4RKD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DNTLMF54N3OHXFOLC6ZNNN4RKD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DNTLMF54N3OHXFOLC6ZNNN4RKD","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a99900fa7e89bce8f474d939c7f9e40e245f3fb5f5fc8122463cb7b09dde4560","cross_cats_sorted":["stat.ME","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-04-21T11:34:24Z","title_canon_sha256":"7da79c1ae9c1e0d0e2b40c64b81e044822be740dac21c4e73b8500e50af00f26"},"schema_version":"1.0","source":{"id":"2604.19353","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.19353","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"arxiv_version","alias_value":"2604.19353v2","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.19353","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"pith_short_12","alias_value":"DNTLMF54N3OH","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"pith_short_16","alias_value":"DNTLMF54N3OHXFOL","created_at":"2026-05-25T02:02:15Z"},{"alias_kind":"pith_short_8","alias_value":"DNTLMF54","created_at":"2026-05-25T02:02:15Z"}],"graph_snapshots":[{"event_id":"sha256:5f7e920799daa177ed2d44aafe1f9303c30d7950b8fa06edfa8c9cbb78f692ae","target":"graph","created_at":"2026-05-25T02:02:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We derive an asymptotic version of Ville's inequality, which bounds excursion probabilities of (E_{m,n})_{m,n∈ℕ} over some threshold uniformly over n up to a time horizon r_m that is determined by the quality of process approximation over m."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The approximation quality of the doubly indexed process to an e-process as m→∞ is sufficient to determine a growing time horizon r_m over which the uniform bound holds, as stated in the abstract's description of the limiting behavior."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Asymptotic e-processes approximate e-processes for large m, enabling an asymptotic Ville's inequality that bounds uniform excursion probabilities up to a time horizon r_m determined by approximation quality."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality."}],"snapshot_sha256":"a0c3954425e721992af4b5cb69ebf01e28da0f94faf70212fee8759adad34d9b"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-20T03:04:23.787679Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.19353/integrity.json","findings":[],"snapshot_sha256":"d29b5909a2e28753a251a729ac5210149a380e4041e23f26abad9d113919e273","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate the concept of an asymptotic e-process, which is a doubly-indexed stochastic process $(E_{m,n})_{m,n\\in\\mathbb{N}}$ that possesses, asymptotically for an approximation index $m\\to\\infty$, the properties of an e-process along a monitoring time index $n$. This constitutes the first in-depth study of this recently introduced concept, which is relevant in asymptotic sequential anytime-valid inference. Our theory is motivated by practical applications in sequential hypothesis testing, in which e-variables and e-processes can only be constructed approximately from observations due to ","authors_text":"Mattes Mollenhauer, Pierre-Fran\\c{c}ois Massiani, Sebastian Schulze","cross_cats":["stat.ME","stat.TH"],"headline":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-04-21T11:34:24Z","title":"Asymptotic e-processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.19353","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T01:34:44.177933Z","id":"53ad64bb-6489-425a-a83f-1c07372d799f","model_set":{"reader":"grok-4.3"},"one_line_summary":"Asymptotic e-processes approximate e-processes for large m, enabling an asymptotic Ville's inequality that bounds uniform excursion probabilities up to a time horizon r_m determined by approximation quality.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A doubly indexed process approximating an e-process satisfies an asymptotic Ville inequality that uniformly bounds its excursions up to a time horizon growing with approximation quality.","strongest_claim":"We derive an asymptotic version of Ville's inequality, which bounds excursion probabilities of (E_{m,n})_{m,n∈ℕ} over some threshold uniformly over n up to a time horizon r_m that is determined by the quality of process approximation over m.","weakest_assumption":"The approximation quality of the doubly indexed process to an e-process as m→∞ is sufficient to determine a growing time horizon r_m over which the uniform bound holds, as stated in the abstract's description of the limiting behavior."}},"verdict_id":"53ad64bb-6489-425a-a83f-1c07372d799f"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:30e1181c1da0be094d15da8d49550b39168b1c28b200025598c34b997f99a771","target":"record","created_at":"2026-05-25T02:02:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a99900fa7e89bce8f474d939c7f9e40e245f3fb5f5fc8122463cb7b09dde4560","cross_cats_sorted":["stat.ME","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2026-04-21T11:34:24Z","title_canon_sha256":"7da79c1ae9c1e0d0e2b40c64b81e044822be740dac21c4e73b8500e50af00f26"},"schema_version":"1.0","source":{"id":"2604.19353","kind":"arxiv","version":2}},"canonical_sha256":"1b66b617bc6edc7b95cb17b2d6b79150f267cf243aa1ab41503caf2eb6ee0aeb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b66b617bc6edc7b95cb17b2d6b79150f267cf243aa1ab41503caf2eb6ee0aeb","first_computed_at":"2026-05-25T02:02:15.578541Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:02:15.578541Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gm0wiIeDyHNIwmwp6sIWYNpYhTOryt0En9O/Yo2mR+gHjwISONcJpCN2Y3ZZjQ8pav2L/LhsyWUU+8qjS84tBA==","signature_status":"signed_v1","signed_at":"2026-05-25T02:02:15.579225Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.19353","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:30e1181c1da0be094d15da8d49550b39168b1c28b200025598c34b997f99a771","sha256:5f7e920799daa177ed2d44aafe1f9303c30d7950b8fa06edfa8c9cbb78f692ae"],"state_sha256":"7b5621b026b2ff78d575d1c4e4c22b4d1c9d3555ddc39aba1b912f6fa23384fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fs+8Dd04chDCOf2j37k8AsaZDrfNv8Jt1nxN+5CkbwP5odQbQH3H7Q3RlxCeALd96GHMLGCzalvO81a1DFdfBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T22:39:19.146795Z","bundle_sha256":"882ba8f0df50e8386ee9da51c0c82e2e21915ea1869ccef59ce8485174040c89"}}