{"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. This result is originally due to Kest"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.1728","kind":"arxiv","version":3},"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"}