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It is proved that $\\displaystyle \\varlimsup_{t \\to \\infty} \\frac{\\|X_t\\|}{\\sqrt{\\log t}} =\\sqrt{2 \\lambda_1}$ almost surely with $\\lambda_1$ being the largest eigenvalue of the matrix $\\displaystyle \\Sigma :=\\int_0^\\infty e^{-s A} \\cdot (D \\cdot D^T) \\cdot e^{-s A^T} \\mathrm{d} s$; the discarded measure-zero set can be chosen independent of the initial values $X_"},"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":"1407.2725","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-07-10T08:54:40Z","cross_cats_sorted":[],"title_canon_sha256":"5bbbc8f6a48ef9ce875443d38f4329cf6f9ed952b2af2361b65f31212ecaa32a","abstract_canon_sha256":"5f66181b1ae27606ab7589fcff3c1fa6be3daebe8aeb3f2a4650395a86bb2dc6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:32.612201Z","signature_b64":"umq69MBBDBqnfOUB+jmwHmBtnw9LUAKqFFyLDLB9hoqyX6E+WTAJwd4BMnPsE7eudNOwLLxGA9hiYCjwkFxeAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1ad5240e81abdc5608ad31fb9901e6ed4b3e6808893ca6176df9e1f7be3fad0","last_reissued_at":"2026-05-18T02:47:32.611547Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:32.611547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounding Ornstein-Uhlenbeck Processes and Alikes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jian-Sheng Xie","submitted_at":"2014-07-10T08:54:40Z","abstract_excerpt":"In this note we consider SDEs of the type $\\mathrm{d} X_t=[F (X_t) -A X_t] \\mathrm{d} t +D \\mathrm{d} W_t$ under the assumptions that $A$'s eigenvalues are all of positive real parts and $F (\\cdot)$ has slower-than-linear growth rate. 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