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Under suitable assumptions, this equation has a unique stationary solution $R$ such that, for some $\\kappa>0$ and some finite positive and continuous function $K$ on $S^{d-1}:=\\{x \\in \\mathbb{R}^{d}:|x|=1\\}$, $ \\lim_{t \\to \\infty} t^{\\kappa} P(xR>t)=K(x)$ for all $x \\in S^{d-1} $ holds true. This result is originally due to Kest"},"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":"1009.1728","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-09-09T10:20:39Z","cross_cats_sorted":[],"title_canon_sha256":"1cedc6e347127faec98175d88813d83c0fb21d5af387b187bd7c49374d4cedaa","abstract_canon_sha256":"f36532fef160d32dc2d5a80c24e7e48f47fe1267a8507c8aeb41d54df210fcf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:56.561705Z","signature_b64":"CQzrMr5xPsW9n08VXeCNwpek0vWE1vrhAenpbvwFAdGq70Fu6LOxo2mUV8HxSAh0Z2G+AHJd+rPcnwIkGB6eCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6fe518b7e18e9c602cc48e9f5f55dce89e0b49a33fadd79cf36f14cc3fbded3","last_reissued_at":"2026-05-18T03:28:56.561013Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:56.561013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tail behavior of stationary solutions of random difference equations: the case of regular matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gerold Alsmeyer, Sebastian Mentemeier","submitted_at":"2010-09-09T10:20:39Z","abstract_excerpt":"Given a sequence $(M_{n},Q_{n})_{n\\ge 1}$ of i.i.d. random variables with generic copy $(M,Q)$ such that $M$ is a regular $d\\times d$ matrix and $Q$ takes values in $\\mathbb{R}^{d}$, we consider the random difference equation (RDE) $R_{n}=M_{n}R_{n-1}+Q_{n}$, $n\\ge 1$. Under suitable assumptions, this equation has a unique stationary solution $R$ such that, for some $\\kappa>0$ and some finite positive and continuous function $K$ on $S^{d-1}:=\\{x \\in \\mathbb{R}^{d}:|x|=1\\}$, $ \\lim_{t \\to \\infty} t^{\\kappa} P(xR>t)=K(x)$ for all $x \\in S^{d-1} $ holds true. 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