{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ONKDFIAOLUY7UAWMZRHUHLAGZO","short_pith_number":"pith:ONKDFIAO","canonical_record":{"source":{"id":"1907.05833","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-12T16:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"2d302f3d5ebd1e2666857b5e7a6f1e2dfda9517ba02f73240556bbcee9861dda","abstract_canon_sha256":"9b0ab20ccc05e644c350a013d245c14d743450fe5c3088463dd5d4688518bbd8"},"schema_version":"1.0"},"canonical_sha256":"735432a00e5d31fa02cccc4f43ac06cb9b5d3cc9bf1c4f273fa12c2562cc78cb","source":{"kind":"arxiv","id":"1907.05833","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.05833","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"arxiv_version","alias_value":"1907.05833v1","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.05833","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"pith_short_12","alias_value":"ONKDFIAOLUY7","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"ONKDFIAOLUY7UAWM","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"ONKDFIAO","created_at":"2026-05-18T12:33:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ONKDFIAOLUY7UAWMZRHUHLAGZO","target":"record","payload":{"canonical_record":{"source":{"id":"1907.05833","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-12T16:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"2d302f3d5ebd1e2666857b5e7a6f1e2dfda9517ba02f73240556bbcee9861dda","abstract_canon_sha256":"9b0ab20ccc05e644c350a013d245c14d743450fe5c3088463dd5d4688518bbd8"},"schema_version":"1.0"},"canonical_sha256":"735432a00e5d31fa02cccc4f43ac06cb9b5d3cc9bf1c4f273fa12c2562cc78cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:44.260718Z","signature_b64":"MqBvNAhjRV9rNmJzdbWpAU3qlxAOPhyc05L05LkY5h2G7l5d92C3iO9llrYujlLGVCdFzPqFmfiRlCBKTrXIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"735432a00e5d31fa02cccc4f43ac06cb9b5d3cc9bf1c4f273fa12c2562cc78cb","last_reissued_at":"2026-05-17T23:40:44.259976Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:44.259976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.05833","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JBjuZtqOA3v+BhETFOG7LmdaQrAC9MNiEvgqTo5Q27qJz4ebbEW6SwZw01dzBUuMhX6xit1DUe7c+Y3wPQQ9AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T18:08:49.971116Z"},"content_sha256":"2d2d2598040edf8c60daa0818787daf12228eabd78eb53422bf02eb2a2bf1e92","schema_version":"1.0","event_id":"sha256:2d2d2598040edf8c60daa0818787daf12228eabd78eb53422bf02eb2a2bf1e92"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ONKDFIAOLUY7UAWMZRHUHLAGZO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Concentration inequalities for random matrix products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Amelia Henriksen, Rachel Ward","submitted_at":"2019-07-12T16:50:36Z","abstract_excerpt":"Suppose $\\{ X_k \\}_{k \\in \\mathbb{Z}}$ is a sequence of bounded independent random matrices with common dimension $d\\times d$ and common expectation $\\mathbb{E}[ X_k ]= X$. Under these general assumptions, the normalized random matrix product $$Z_n = (I + \\frac{1}{n}X_n)(I + \\frac{1}{n}X_{n-1}) \\cdots (I + \\frac{1}{n}X_1)$$ converges to $Z_n \\rightarrow e^{X}$ as $n \\rightarrow \\infty$. Normalized random matrix products of this form arise naturally in stochastic iterative algorithms, such as Oja's algorithm for streaming Principal Component Analysis. Here, we derive nonasymptotic concentration"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05833","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"klcPez1ABrYLCMCpZo/72gOQLgJ1PD9C81hg0MKhnECNF4nuTATIZ7pnv8kOM8HbRT8Frlla5IFEv6+MOLEEAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T18:08:49.971465Z"},"content_sha256":"8bb1138bf662ed497595ab090b805ae6c060e7a1174b56ea4befa5aa19f6bd7d","schema_version":"1.0","event_id":"sha256:8bb1138bf662ed497595ab090b805ae6c060e7a1174b56ea4befa5aa19f6bd7d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ONKDFIAOLUY7UAWMZRHUHLAGZO/bundle.json","state_url":"https://pith.science/pith/ONKDFIAOLUY7UAWMZRHUHLAGZO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ONKDFIAOLUY7UAWMZRHUHLAGZO/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":"2026-06-28T18:08:49Z","links":{"resolver":"https://pith.science/pith/ONKDFIAOLUY7UAWMZRHUHLAGZO","bundle":"https://pith.science/pith/ONKDFIAOLUY7UAWMZRHUHLAGZO/bundle.json","state":"https://pith.science/pith/ONKDFIAOLUY7UAWMZRHUHLAGZO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ONKDFIAOLUY7UAWMZRHUHLAGZO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ONKDFIAOLUY7UAWMZRHUHLAGZO","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":"9b0ab20ccc05e644c350a013d245c14d743450fe5c3088463dd5d4688518bbd8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-12T16:50:36Z","title_canon_sha256":"2d302f3d5ebd1e2666857b5e7a6f1e2dfda9517ba02f73240556bbcee9861dda"},"schema_version":"1.0","source":{"id":"1907.05833","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.05833","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"arxiv_version","alias_value":"1907.05833v1","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.05833","created_at":"2026-05-17T23:40:44Z"},{"alias_kind":"pith_short_12","alias_value":"ONKDFIAOLUY7","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"ONKDFIAOLUY7UAWM","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"ONKDFIAO","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:8bb1138bf662ed497595ab090b805ae6c060e7a1174b56ea4befa5aa19f6bd7d","target":"graph","created_at":"2026-05-17T23:40:44Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Suppose $\\{ X_k \\}_{k \\in \\mathbb{Z}}$ is a sequence of bounded independent random matrices with common dimension $d\\times d$ and common expectation $\\mathbb{E}[ X_k ]= X$. Under these general assumptions, the normalized random matrix product $$Z_n = (I + \\frac{1}{n}X_n)(I + \\frac{1}{n}X_{n-1}) \\cdots (I + \\frac{1}{n}X_1)$$ converges to $Z_n \\rightarrow e^{X}$ as $n \\rightarrow \\infty$. Normalized random matrix products of this form arise naturally in stochastic iterative algorithms, such as Oja's algorithm for streaming Principal Component Analysis. Here, we derive nonasymptotic concentration","authors_text":"Amelia Henriksen, Rachel Ward","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-12T16:50:36Z","title":"Concentration inequalities for random matrix products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05833","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2d2d2598040edf8c60daa0818787daf12228eabd78eb53422bf02eb2a2bf1e92","target":"record","created_at":"2026-05-17T23:40:44Z","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":"9b0ab20ccc05e644c350a013d245c14d743450fe5c3088463dd5d4688518bbd8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-12T16:50:36Z","title_canon_sha256":"2d302f3d5ebd1e2666857b5e7a6f1e2dfda9517ba02f73240556bbcee9861dda"},"schema_version":"1.0","source":{"id":"1907.05833","kind":"arxiv","version":1}},"canonical_sha256":"735432a00e5d31fa02cccc4f43ac06cb9b5d3cc9bf1c4f273fa12c2562cc78cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"735432a00e5d31fa02cccc4f43ac06cb9b5d3cc9bf1c4f273fa12c2562cc78cb","first_computed_at":"2026-05-17T23:40:44.259976Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:44.259976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MqBvNAhjRV9rNmJzdbWpAU3qlxAOPhyc05L05LkY5h2G7l5d92C3iO9llrYujlLGVCdFzPqFmfiRlCBKTrXIAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:44.260718Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.05833","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d2d2598040edf8c60daa0818787daf12228eabd78eb53422bf02eb2a2bf1e92","sha256:8bb1138bf662ed497595ab090b805ae6c060e7a1174b56ea4befa5aa19f6bd7d"],"state_sha256":"7bb3b9aeb388efc2587ce957da0af214379902138320727921fb3af0205eddfc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xmk77VaHav+6IwkVo3/qEHgx7I6MXLMkGcYOrbGhrVcK2+rTMPXbF3hG42AMJulPhZemZlmMBL7jICuLIStOAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T18:08:49.973306Z","bundle_sha256":"7b1ac3fa57e572df68a4574a5759f91ebf6df6c8099b3f4c800a92b59eb3f0e2"}}