{"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 $\\kappa$. As a by-product, we show that the unique stationary solution $R$ of the R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0797","kind":"arxiv","version":2},"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"}