{"paper":{"title":"Strong renewal theorems and local large deviations for multivariate random walks and renewals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Quentin Berger","submitted_at":"2018-07-10T11:38:34Z","abstract_excerpt":"We study a random walk $\\mathbf{S}_n$ on $\\mathbb{Z}^d$ ($d\\geq 1$), in the domain of attraction of an operator-stable distribution with index $\\boldsymbol{\\alpha}=(\\alpha_1,\\ldots,\\alpha_d) \\in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function $G(\\mathbf{0},\\mathbf{x})$ as $\\|\\mathbf{x}\\|\\to +\\infty$, along the \"favorite direction or scaling\": (i) if $\\sum_{i=1}^d \\alpha_i^{-1} < 2$ (reminiscent of Garsia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\\mathbf{3"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03575","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"}