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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"},"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":"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"},"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"},"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 $\\mathbb{E}[ X_k ]= X$. 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