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M_n}^\\kappa})^{\\frac{1}{n}} = 1 .$$ We prove, under suitable assumptions, that the sequence $S_n = R_1 + ... + R_n$, appropriately normalized, converges in law to a multidimensional stable distribution with index $\\kappa$. As a by-product, we show that the unique stationary solution $R$ of the R"},"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":"1105.0797","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-05-04T11:55:20Z","cross_cats_sorted":[],"title_canon_sha256":"ae651db0bea00d96dec32302ae3000954ef93bc2cfb673497fd9c526a2f84dc0","abstract_canon_sha256":"de3ffa935ab00180dea9a372255881c653674a7171c1cf47f0349b6c3a46fe5c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:14.115889Z","signature_b64":"OEO+Ml6KIhi8pj6uka4zmRD9b87FoNXAcuCq8T7aSaSzQTrdWX9smaAgO1XZUmSb45+GE5KolxN+x866blTfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59411c383be94d505e0100fcd41b956211c715eb388d21b8338c03b57662c40c","last_reissued_at":"2026-05-18T03:29:14.115241Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:14.115241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ewa Damek, Jacek Zienkiewicz, Mariusz Mirek, Sebastian Mentemeier","submitted_at":"2011-05-04T11:55:20Z","abstract_excerpt":"Given a sequence $(M_{n},Q_{n})_{n\\ge 1}$ of i.i.d.\\ random variables with generic copy $(M,Q) \\in GL(d, \\R) \\times \\R^d$, we consider the random difference equation (RDE) $$ R_{n}=M_{n}R_{n-1}+Q_{n}, $$ $n\\ge 1$, and assume the existence of $\\kappa >0$ such that $$ \\lim_{n \\to \\infty}(\\E{\\norm{M_1 ... M_n}^\\kappa})^{\\frac{1}{n}} = 1 .$$ We prove, under suitable assumptions, that the sequence $S_n = R_1 + ... + R_n$, appropriately normalized, converges in law to a multidimensional stable distribution with index $\\kappa$. 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