{"paper":{"title":"Moment convergence of first-passage times in renewal theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Iksanov, Alexander Marynych, Matthias Meiners","submitted_at":"2012-08-20T10:27:08Z","abstract_excerpt":"Let $\\xi_1, \\xi_2, \\ldots$ be independent copies of a positive random variable $\\xi$, $S_0 = 0$, and $S_k = \\xi_1+\\ldots+\\xi_k$, $k \\in \\mathbb{N}$. Define $N(t) = \\inf\\{k \\in \\mathbb{N}: S_k>t\\}$ for $t\\geq 0$. The process $(N(t))_{t\\geq 0}$ is the first-passage time process associated with $(S_k)_{k\\geq 0}$. It is known that if the law of $\\xi$ belongs to the domain of attraction of a stable law or $\\mathbb{P}(\\xi>t)$ varies slowly at $\\infty$, then $N(t)$, suitably shifted and scaled, converges in distribution as $t \\to \\infty$ to a random variable $W$ with a stable law or a Mittag-Leffler "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.3964","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"}