{"paper":{"title":"Local asymptotics for the first intersection of two independent renewals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Kenneth S. Alexander, Quentin Berger","submitted_at":"2016-03-17T15:13:26Z","abstract_excerpt":"We study the intersection of two independent renewal processes, $\\rho=\\tau\\cap\\sigma$. Assuming that $\\mathbf{P}(\\tau_1 = n ) = \\varphi(n)\\, n^{-(1+\\alpha)}$ and $\\mathbf{P}(\\sigma_1 = n ) = \\tilde\\varphi(n)\\, n^{-(1+ \\tilde\\alpha)} $ for some $\\alpha,\\tilde \\alpha \\geq 0$ and some slowly varying $\\varphi,\\tilde\\varphi$, we give the asymptotic behavior first of $\\mathbf{P}(\\rho_1>n)$ (which is straightforward except in the case of $\\min(\\alpha,\\tilde\\alpha)=1$) and then of $\\mathbf{P}(\\rho_1=n)$. The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities $\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.05531","kind":"arxiv","version":1},"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"}